C₂×#2396 #34543

order	256
nilpotency class	2
isoclinism	isoclinic class of #392
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

	subgroup	G/H
center	C₂³×C₄	C₂³
commutator, Frattini, cosocle	C₂³	C₂⁵
socle	C₂⁴	C₂⁴
prime radical	C₂⁵	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

1483 conjugacy classes of subgroups.

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

A minimal faithful permutation representation — 5 generators on 22 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 27531);