C₂×#2805710 #72337884

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹¹×C₄²

Aliases

direct product

C₂×#2805710

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

48351 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:

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232289, C₂×#72232334, C₂×#72232311
maximal normal subgroup of	C₂×#72232289, C₂×#72232334, C₂×#72232311

GAP

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

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

C2×#2805710 #72337884