C₂×#7856 #180862

order	512
nilpotency class	2
isoclinism	isoclinic class of #7855
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

direct product

C₂×#7856

Subgroups

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

1890 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);(3,4,6,8,7,9,5,10)(11,12)(13,19,21,25)(14,17)(15,18)(16,20,23,26)(22,24);(11,13)(12,16,18,23)(14,20)(15,21)(17,19,24,25)(22,26);(3,5,7,6)(4,8,9,10)

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#233608, #72216622
maximal normal subgroup of	C₂×#233608, #72216622
maximal quotient of	C₂×#105252, C₂×#94876, C₂×#94877

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88058);

C2×#7856 #180862

Aliases

Subgroups

Presentation

Conjugacy classes & character table

Relationships

GAP

C₂×#7856 #180862