C₂×#8191 #182333

order	512
nilpotency class	3
isoclinism	isoclinic class of AGL(1,ℤ/8)
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

direct product

C₂×#8191

Subgroups

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

1306 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);(3,4,6,9,7,5,8,10)(11,12)(13,19)(14,22)(15,18)(16,21)(17,23)(20,25)(24,26);(11,13)(12,16,18,24)(14,21)(15,20)(17,19,22,25)(23,26);(3,5,6,10,7,4,8,9)(13,20)(16,24)(19,25)(21,26)

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×C₈×#444, C₂×#184011
maximal normal subgroup of	C₂×C₈×#444, C₂×#184011
maximal quotient of	C₂×#96686

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 89529);

C2×#8191 #182333

Aliases

Subgroups

Presentation

Conjugacy classes & character table

Relationships

GAP

C₂×#8191 #182333