C₂×#778050 #72302467

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

Aliases

direct product

C₂×#778050

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂³×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

29149 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)(11,12,15,14)(13,16,17,18)(19,20)(21,24)(22,23)(25,26);(5,6)(8,10)(12,14)(16,18)(20,23)(24,26);(3,5)(4,6)(7,8)(9,10)(13,17)(16,18)(20,23)(24,26);(5,6)(7,8)(9,10)(12,14)(16,18)(19,20,22,23)(21,24,25,26);(5,6)(7,8)(9,10)(11,13,15,17)(12,16,14,18)(19,21,22,25)(20,24,23,26);(3,5)(4,6)(11,14)(12,15)(13,16)(17,18)

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72279950, C₂×#72280075, C₂×#72280294
maximal normal subgroup of	C₂×#72279950, C₂×#72280075, C₂×#72280294

GAP

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

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

C2×#778050 #72302467