  In this paper, we explore the descriptive complexity theory of finite groups
by examining the power of the second Ehrenfeucht-Fraisse bijective pebble game
in Hella's (Ann. Pure Appl. Log., 1989) hierarchy. This is a Spoiler-Duplicator
game in which Spoiler can place up to two pebbles each round. While it
trivially solves graph isomorphism, it may be nontrivial for finite groups, and
other ternary relational structures. We first provide a novel generalization of
Weisfeiler-Leman (WL) coloring, which we call 2-ary WL. We then show that the
2-ary WL is equivalent to the second Ehrenfeucht-Fraisse bijective pebble game
in Hella's hierarchy.
  Our main result is that, in the pebble game characterization, only O(1)
pebbles and O(1) rounds are sufficient to identify all groups without Abelian
normal subgroups (a class of groups for which isomorphism testing is known to
be in P; Babai, Codenotti, & Qiao, ICALP 2012). In particular, we show that
within the first few rounds, Spoiler can force Duplicator to select an
isomorphism between two such groups at each subsequent round. By Hella's
results (ibid.), this is equivalent to saying that these groups are identified
by formulas in first-order logic with generalized 2-ary quantifiers, using only
O(1) variables and O(1) quantifier depth.
