C₂×#7885 #180944

order	512
nilpotency class	3
isoclinism	isoclinic class of #465
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#7885

Subgroups

	subgroup	G/H
center	C₂×C₄²	C₂×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×#95
prime radical	C₂⁵	C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1934 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)(5,7)(6,9)(8,10)(11,12,14,19,16,18,15,20)(13,17,21,25,23,24,22,26);(4,7,9,8)(11,13,15,22,16,23,14,21)(12,17,20,26,18,24,19,25);(12,18)(17,24)(19,20)(25,26)

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#109018, #72212990, #72212999, …
maximal normal subgroup of	C₂×#109018, #72212990, #72212999, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88140);

C2×#7885 #180944

Aliases

Subgroups

Presentation

Conjugacy classes & character table

Relationships

GAP

C₂×#7885 #180944