C₂×#11592 #481055

order	512
nilpotency class	4
isoclinism	isoclinic class of #509
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#11592

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×D₈
commutator	C₈	C₂³×C₈
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂³	C₂×#102
prime radical	C₂²×#135	C₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1084 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 256

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

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×C₈×#509, C₂×#128691, C₂×#128557
maximal normal subgroup of	C₂×C₈×#509, C₂×#128691, C₂×#128557
maximal quotient of	C₂×#128531

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 388251);

C2×#11592 #481055

Aliases

Subgroups

Presentation

Conjugacy classes & character table

Relationships

GAP

C₂×#11592 #481055