C₂×#28573 #6459532

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	C₂²×#466	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3080 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)(9,13)(10,12)(11,14)(16,19)(17,21)(20,22);(3,4,5,6)(7,9,10,14)(8,11,12,13)(16,20)(19,22);(4,6)(15,16,17,19,18,20,21,22);(8,12)(11,13)

maximal subgroup of	C₂×#6596621, C₂×#6599692, C₂×#6468810
maximal normal subgroup of	C₂×#6596621, C₂×#6599692, C₂×#6468810

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6366728);