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C₂×C₈×#108 #183500

order	512
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

C₂×C₈×#108

Subgroups

	subgroup	G/H
center	C₂×C₄×C₈	D₄
commutator	C₄	C₂²×C₄×C₈
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₄×C₄⋊C₄
prime radical	C₂⁴	C₄×Q₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₈ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

902 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 256

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

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

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×C₈×#365, C₂×C₈×#444, C₂×C₈×#361, …
maximal normal subgroup of	C₂×C₈×#365, C₂×C₈×#444, C₂×C₈×#361, …
maximal quotient of	C₂×C₈×#334

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 90696);