C₂×#25955 #6438053

order	512
nilpotency class	3
isoclinism	isoclinic class of #23848
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3070 conjugacy classes of subgroups.

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

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

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

maximal subgroup of	C₂×#6439861, C₂×#7518605, C₂×#7517084
maximal normal subgroup of	C₂×#6439861, C₂×#7518605, C₂×#7517084

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6345249);