C₂×#24215 #6442501

order	512
nilpotency class	3
isoclinism	isoclinic class of #23918
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#447	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

5718 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);(4,5)(11,12)(13,15)(14,16)(17,18);(3,4,6,5)(7,8)(9,10)(12,15)(16,17);(8,10)(12,16)(14,18)(15,17);(3,5,6,4)(12,15)(16,17)

maximal subgroup of	#72310181, C₂×#6453730, C₂×D₄×#463, …
maximal normal subgroup of	#72310181, C₂×#6453730, C₂×D₄×#463, …

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6349697);