C₂×#9089 #196075

order	512
nilpotency class	3
isoclinism	isoclinic class of #1902
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁷

Aliases

direct product

C₂×#9089

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂³×D₈	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1598 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)(11,12,17,20)(13,16,19,15)(14,21)(18,22)(23,25)(24,26);(3,5,6,10,7,4,8,9)(11,13,15,12,17,19,16,20)(14,21,23,18,24,26,22,25);(11,14)(12,18)(13,21)(15,23)(16,22)(17,24)(19,26)(20,25)

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#196123, #72237150, #72238934, …
maximal normal subgroup of	C₂×#196123, #72237150, #72238934, …
maximal quotient of	#72237150, C₂²×#9089, #72309103, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 103271);

C2×#9089 #196075

Aliases

Subgroups

Presentation

Conjugacy classes & character table

Relationships

GAP

C₂×#9089 #196075