C₂×#2807696 #72337895

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²×C₄

Aliases

direct product

C₂×#2807696

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

44425 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:

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232354, C₂×#72232334, C₂×#72232296
maximal normal subgroup of	C₂×#72232354, C₂×#72232334, C₂×#72232296

GAP

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

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

C2×#2807696 #72337895