C₂×#24838 #6445605

order	512
nilpotency class	3
isoclinism	isoclinic class of #24378
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

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

2670 conjugacy classes of subgroups.

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 22 points:

(1,2);(7,8,11,14)(9,13,10,12)(15,16,19,22,18,21,17,20);(3,4)(5,6)(7,9,11,10)(8,12,14,13);(4,6)(15,16,18,21)(17,22,19,20);(7,10,11,9)(8,13,14,12)

maximal subgroup of	C₂×#7520857, C₂×#7522138, C₂×#7518310, …
maximal normal subgroup of	C₂×#7520857, C₂×#7522138, C₂×#7518310, …

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6352801);