C₂×#8294 #183501

order	512
nilpotency class	3
isoclinism	isoclinic class of AGL(1,ℤ/8)
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₂×#8294

Subgroups

	subgroup	G/H
center	C₂×C₄²	C₂×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×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

862 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);(3,4,6,9,7,5,8,10)(12,17,19,23)(13,20)(18,25,26,24)(21,22);(11,12,15,17,16,19,14,23)(13,18,22,24,20,26,21,25);(3,5,6,10,7,4,8,9)(11,13,14,21,16,20,15,22)(12,18,23,25,19,26,17,24)

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#129171, C₂×#129166, C₂×C₈×#444, …
maximal normal subgroup of	C₂×#129171, C₂×#129166, C₂×C₈×#444, …
maximal quotient of	C₂×#96892

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 90697);

C2×#8294 #183501

Aliases

Subgroups

Presentation

Conjugacy classes & character table

Relationships

GAP

C₂×#8294 #183501