C₂×#2088 #23664

order	256
nilpotency class	3
isoclinism	isoclinic class of #1999
automorphism	Aut(G) order 131072
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

527 conjugacy classes of subgroups.

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 256

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 16652);