C₂×#726188 #72302650

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴

Aliases

direct product

C₂×#726188

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

56985 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 256

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

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72280046, C₂×#72279893, C₂×#72280470, …
maximal normal subgroup of	C₂×#72280046, C₂×#72279893, C₂×#72280470, …

GAP

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

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

C2×#726188 #72302650