C₂²×#3094 #10575026

order	512
nilpotency class	3
isoclinism	isoclinic class of #3003
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁶
prime radical	C₂³×#121	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂⁵×C₄ ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

10302 conjugacy classes of subgroups.

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 20 points:

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

maximal subgroup of	C₂×#10578014, C₂²×#60750, C₂×#6440417, …
maximal normal subgroup of	C₂×#10578014, C₂²×#60750, C₂×#6440417, …

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10482222);