C₂×#23454 #6431804

order	512
nilpotency class	3
isoclinism	isoclinic class of #23391
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	C₂⁴×D₄	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

5094 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);(7,8,10,12)(9,13,14,11)(15,16)(17,21)(18,20)(19,22);(3,4)(5,6)(7,9,10,14)(8,11,12,13);(4,6)(16,20)(17,18)(21,22);(15,17,19,18)(16,21,22,20)

maximal subgroup of	C₂×#6451831, C₂×#6463609, C₂×#6432661, …
maximal normal subgroup of	C₂×#6451831, C₂×#6463609, C₂×#6432661, …

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6339000);