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C₂²×#2163 #7622543

order	512
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₈ · Schur multiplier (H₂) = C₂⁹

Aliases

Subgroups

	subgroup	G/H
center	C₂³×C₈	C₂³
commutator	C₂²	C₂⁴×C₈
Frattini, cosocle	C₂×C₈	C₂⁵
socle	C₂⁴	C₂²×C₈
prime radical	C₂⁵	C₂×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2018 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

(1,2);(3,4);(5,6)(7,10)(8,9)(11,12)(13,14,15,19,16,20,21,26,17,18,22,24,23,25,27,28);(6,9)(10,12);(5,7,8,11)(6,10,9,12)(14,18)(19,24)(20,25)(26,28)

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#11654, C₂²×#12030, C₂²×#33669, …
maximal normal subgroup of	C₂²×#11654, C₂²×#12030, C₂²×#33669, …
maximal quotient of	C₂×#512736, C₂²×#12599

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7529739);