C₂×#10707 #353143

order	512
nilpotency class	3
isoclinism	isoclinic class of AGL(1,ℤ/8)
automorphism	?
homology	Abelianization (H₁) = C₂×C₄³ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

direct product

C₂×#10707

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂×D₄
commutator	C₄	C₂×C₄³
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂⁴	C₄×D₄
prime radical	C₂⁵	#41
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ C₂⁴×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2618 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)(5,6)(7,8,9,10)(11,12)(13,18)(14,17)(15,19)(16,20)(21,26)(22,24)(23,25);(4,6)(12,17,20,19)(13,21)(18,25,26,24)(22,23);(11,13,15,23,16,21,14,22)(12,18,19,25,20,26,17,24)

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#354908, C₂×#354302, C₂×#359125, …
maximal normal subgroup of	C₂×#354908, C₂×#354302, C₂×#359125, …
maximal quotient of	C₂×#105326, #72318216, #72321675

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 260339);

C2×#10707 #353143

Aliases

Subgroups

Presentation

Conjugacy classes & character table

Relationships

GAP

C₂×#10707 #353143