C₂×#27645 #6457189

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×#121	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3054 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);(4,5)(7,8,10,12)(9,11,13,14)(15,16,18,21)(17,19,22,20);(3,4,6,5)(7,9)(8,11)(10,13)(12,14);(9,13)(11,14)(16,19)(17,22)(20,21);(3,5,6,4)

maximal subgroup of	C₂×#6578956, C₂×#6467144, C₂×D₄×#481, …
maximal normal subgroup of	C₂×#6578956, C₂×#6467144, C₂×D₄×#481, …

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6364385);