C₂×#1506686 #72337825

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

Aliases

direct product

C₂×#1506686

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

50629 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)(12,15)(16,17)(19,20)(21,24)(22,23)(25,26);(7,8,9,10)(11,12,14,15)(13,17,18,16)(19,21,23,25)(20,24,22,26);(8,10)(13,18)(16,17)(20,22)(24,26);(3,4)(5,6)(7,8)(9,10)(11,13)(12,16)(14,18)(15,17)(19,22,23,20)(21,24,25,26);(1,2);(11,12)(13,16)(14,15)(17,18)(19,21)(20,24)(22,26)(23,25)

Conjugacy classes & character table

148 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232281, C₂×#72232354, C₂×#72232322, …
maximal normal subgroup of	C₂×#72232281, C₂×#72232354, C₂×#72232322, …

GAP

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

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

C2×#1506686 #72337825