C₂×#1265714 #72338014

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³×C₄

Aliases

direct product

C₂×#1265714

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

56579 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:

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

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72210496, C₂×#72232330, C₂×#72232298
maximal normal subgroup of	C₂×#72210496, C₂×#72232330, C₂×#72232298

GAP

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

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

C2×#1265714 #72338014