C₂⁶×D₈ #72304942

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁸ · Schur multiplier (H₂) = C₂²⁸

Aliases

direct product

C₂⁶×D₈

Subgroups

	subgroup	G/H
center, socle	C₂⁷	D₄
commutator, Frattini, cosocle	C₄	C₂⁸
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁷ ◁ C₂⁶×C₄ ◁ G
LECS	G ▷ C₄ ▷ C₂ ▷ C₁

1193173 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 8 generators on 20 points:

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

Conjugacy classes & character table

448 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂⁵×#442, C₂×#72237129, C₂×#72231434, …
maximal normal subgroup of	C₂⁵×#442, C₂×#72237129, C₂×#72231434, …
maximal quotient of	C₂⁶×#102, C₂⁷×D₈

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((1,2), (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(14,17)(15,19)(18,20), (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,20)(16,18)(17,19), (3,4), (5,6), (7,8), (9,10), (11,12));

C26×D8 #72304942