C₂×#7745076 #72302201

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁸

Aliases

direct product

C₂×#7745076

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂⁶
commutator, Frattini, cosocle	C₂³	C₂⁷
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂³ ▷ C₁

58627 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 7 generators on 26 points:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72279634
maximal normal subgroup of	C₂×#72279634
maximal quotient of	C₂×#72279893, C₂×#72280033, C₂×#72279963, …

GAP

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

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

C2×#7745076 #72302201