C₂×#2246 #33709

order	256
nilpotency class	4
isoclinism	isoclinic class of #2215
automorphism	Aut(G) order 524288
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

	subgroup	G/H
center, socle	C₂³	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

511 conjugacy classes of subgroups.

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 22 points:

(1,2);(4,6)(8,12)(9,13)(10,16)(14,22)(15,20)(17,21)(18,19);(7,8,13,19,10,14,17,20,11,15,21,22,16,18,9,12);(3,4,5,6)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 26697);