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C₂×#7861 #180878

order	512
nilpotency class	2
isoclinism	isoclinic class of #7855
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×C₄²
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×#95
prime radical	C₂⁵	C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1806 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 26 points:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#232766, C₂×#196123, #72237152, …
maximal normal subgroup of	C₂×#232766, C₂×#196123, #72237152, …
maximal quotient of	C₂×#105255, C₂×#94876, C₂×#94877, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88074);