C₂×#24709 #6449888

order	512
nilpotency class	3
isoclinism	isoclinic class of #24384
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2702 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);(7,8,10,11)(9,13,14,12)(15,16,18,21,17,19,22,20);(3,4)(5,6)(8,11)(12,13)(16,19)(20,21);(4,6)(16,20)(18,22)(19,21);(7,9,10,14)(8,12,11,13)

maximal subgroup of	C₂×#7519680, C₂×#7520299, C₂×#7522896, …
maximal normal subgroup of	C₂×#7519680, C₂×#7520299, C₂×#7522896, …

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6357084);