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C₂×C₈×SD₃₂ #483493

order	512
nilpotency class	4
isoclinism	isoclinic class of D₁₆
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

C₂×C₈×SD₃₂

Subgroups

	subgroup	G/H
center	C₂²×C₈	D₈
commutator	C₈	C₂³×C₈
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂³	C₄×D₈
prime radical	C₂²×D₈	C₂×C₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₈² ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

558 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);(3,4,5,6,7,8,9,10);(11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26);(12,18)(13,25)(14,16)(15,23)(17,21)(20,26)(22,24)

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×C₈×#509, C₂×C₈×#508
maximal normal subgroup of	C₂×C₈×#509, C₂×C₈×#508
maximal quotient of	C₂×C₈×#357, C₂×C₈×#367

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390689);