C₂×#7932 #181094

order	512
nilpotency class	3
isoclinism	isoclinic class of #99
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂⁷

Aliases

direct product

C₂×#7932

Subgroups

	subgroup	G/H
center	C₂×C₄²	C₂²⋊C₄
commutator	C₂²	C₂²×C₄×C₈
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×#95
prime radical	C₂⁵	C₄²
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1926 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);(3,4,6,9,8,10,7,5)(11,12)(13,21,23,17)(14,18)(15,20)(16,19)(22,26,24,25);(3,5,7,10,8,9,6,4)(11,13,15,24,16,23,14,22)(12,17,20,26,19,21,18,25);(11,14,16,15)(12,18,19,20)(13,22,23,24)(17,25,21,26)

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#196174, C₂×#196180, C₂×#181106
maximal normal subgroup of	C₂×#196174, C₂×#196180, C₂×#181106
maximal quotient of	#72309243

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88290);

C2×#7932 #181094

Aliases

Subgroups

Presentation

Conjugacy classes & character table

Relationships

GAP

C₂×#7932 #181094