C₂×#498624 #72314061

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

direct product

C₂×#498624

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#442
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₂²×C₈	C₂⁵
prime radical	C₂²×#493	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂×#1770 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3479 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 26 points:

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

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72265085, C₂×#72264898, C₂×#72264870
maximal normal subgroup of	C₂×#72265085, C₂×#72264898, C₂×#72264870

GAP

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

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

C2×#498624 #72314061