C₂×#795496 #72302704

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

direct product

C₂×#795496

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

33867 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 6 generators on 26 points:

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

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72280460, C₂×#72280294, C₂×#72280221
maximal normal subgroup of	C₂×#72280460, C₂×#72280294, C₂×#72280221

GAP

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

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

C2×#795496 #72302704