C₂×#9874 #206783

order	512
nilpotency class	3
isoclinism	isoclinic class of #1999
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#9874

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle, prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1346 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 26 points:

(1,2);(3,4)(5,9)(6,7)(8,10)(12,15)(13,17)(16,18)(19,20,24,25,23,26,22,21);(4,7)(8,9)(11,12,13,15,14,16,17,18)(19,21,22,26,23,25,24,20);(3,5,6,10)(4,8,7,9)(19,22,23,24)(20,25,26,21)

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#102868, C₂×#207565, C₂×#102858
maximal normal subgroup of	C₂×#102868, C₂×#207565, C₂×#102858
maximal quotient of	C₂×#108795, C₂×#141919, C₂×#141918

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 113979);

C2×#9874 #206783

Aliases

Subgroups

Presentation

Conjugacy classes & character table

Relationships

GAP

C₂×#9874 #206783