C₂×#8184 #182290

order	512
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

direct product

C₂×#8184

Subgroups

	subgroup	G/H
center	C₂×C₄×C₈	D₄
commutator	C₄	C₂²×C₄×C₈
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₄×C₂²⋊C₄
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₈ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1434 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);(11,12,15,19,16,20,14,17)(13,18,22,25,23,26,21,24);(3,4,5,7,6,8,9,10)(12,17,20,19)(18,24,26,25);(11,13,15,22,16,23,14,21)(12,18,19,25,20,26,17,24)

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#184011, C₂×C₈×#443, C₂×#128250
maximal normal subgroup of	C₂×#184011, C₂×C₈×#443, C₂×#128250
maximal quotient of	C₂×C₈×D₄⋊C₈

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 89486);

C2×#8184 #182290

Aliases

Subgroups

Presentation

Conjugacy classes & character table

Relationships

GAP

C₂×#8184 #182290