C₂×#24754 #6442965

order	512
nilpotency class	3
isoclinism	isoclinic class of #24378
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 ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3352 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)(11,12,15,16,14,18,13,17);(3,4,6,5)(7,8)(9,10)(11,13,14,15)(12,16,18,17);(8,10)(12,17)(13,15)(16,18);(3,5,6,4)(12,18)(16,17)

maximal subgroup of	C₂×#7519107, C₂×#7522922, C₂×#7521690, …
maximal normal subgroup of	C₂×#7519107, C₂×#7522922, C₂×#7521690, …

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6350161);