C₂×#25774 #6443198

order	512
nilpotency class	3
isoclinism	isoclinic class of #24264
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#2433	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

6116 conjugacy classes of subgroups.

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 18 points:

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

maximal subgroup of	C₂×SD₁₆×#120, C₂×#6443244, C₂×#6573500, …
maximal normal subgroup of	C₂×SD₁₆×#120, C₂×#6443244, C₂×#6573500, …

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6350394);