C₂×#795112 #72302702

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

Aliases

direct product

C₂×#795112

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂²×#2397	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

24967 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);(3,4,5,6)(19,20,23,21)(22,25,26,24);(7,8)(9,10)(12,15)(13,17)(19,20,23,21)(22,24,26,25);(8,10)(19,21,23,20)(22,25,26,24);(3,4)(5,6)(7,8)(9,10)(11,12,14,15)(13,18,17,16);(3,4)(5,6)(7,8)(9,10)(11,13,14,17)(12,16,15,18)(19,22)(20,24)(21,25)(23,26)

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72280460, C₂×#72280169, C₂×#72280272
maximal normal subgroup of	C₂×#72280460, C₂×#72280169, C₂×#72280272

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

C2×#795112 #72302702