C₂×#28253 #6465469

order	512
nilpotency class	3
isoclinism	isoclinic class of #23918
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×#459	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3450 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);(4,5)(12,15)(13,17)(16,18);(3,4,6,5)(7,8,9,10)(12,16)(15,18);(8,10)(11,12)(13,18)(14,16)(15,17);(3,5,6,4)

maximal subgroup of	C₂×D₄×#463, C₂×#6467137, C₂×#6577868
maximal normal subgroup of	C₂×D₄×#463, C₂×#6467137, C₂×#6577868

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6372665);