C₂×#29712 #6432962

order	512
nilpotency class	3
isoclinism	isoclinic class of #29592
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

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

4466 conjugacy classes of subgroups.

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 128

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

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

maximal subgroup of	C₂×#6674806, C₂×#6481774, C₂×#6668300, …
maximal normal subgroup of	C₂×#6674806, C₂×#6481774, C₂×#6668300, …

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6340158);