C₂×#25090 #6438737

order	512
nilpotency class	3
isoclinism	isoclinic class of #23848
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#466	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3768 conjugacy classes of subgroups.

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 256

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

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

maximal subgroup of	C₂×D₄×#463, C₂×#6439865, C₂×#6577350, …
maximal normal subgroup of	C₂×D₄×#463, C₂×#6439865, C₂×#6577350, …

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6345933);