C₂×#8268 #183335

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

Aliases

direct product

C₂×#8268

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	#117
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

950 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);(12,16,18,21)(13,19)(17,24,25,23)(20,26);(3,4,5,7,6,8,9,10)(11,12)(13,17)(14,21)(15,18)(16,22)(19,25)(20,24)(23,26);(11,13,14,20,15,19,22,26)(12,17,21,24,18,25,16,23)

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#183351, C₂×#128555, C₂×#129120, …
maximal normal subgroup of	C₂×#183351, C₂×#128555, C₂×#129120, …
maximal quotient of	C₂×#96892

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 90531);

C2×#8268 #183335

Aliases

Subgroups

Presentation

Conjugacy classes & character table

Relationships

GAP

C₂×#8268 #183335