C₂×#498464 #72313938

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

direct product

C₂×#498464

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#442
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₂²×C₈	C₂⁵
prime radical	C₂×#33582	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂×#1770 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3745 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)(5,9)(6,8)(7,10)(12,16,19,20)(13,21)(17,24,22,25)(18,26);(1,2);(12,17,19,22)(13,18)(14,23)(16,24,20,25)(21,26);(3,5,6,10)(4,7,8,9);(4,8)(7,9)(11,12,18,25,23,16,13,22,15,19,26,24,14,20,21,17)

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72264735, C₂×#72265085, C₂×#72264870
maximal normal subgroup of	C₂×#72264735, C₂×#72265085, C₂×#72264870

GAP

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

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

C2×#498464 #72313938