C₂×#1515165 #72337873

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

Aliases

direct product

C₂×#1515165

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

46517 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:

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

Conjugacy classes & character table

148 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232354, C₂×#72232311, C₂×#72232330
maximal normal subgroup of	C₂×#72232354, C₂×#72232311, C₂×#72232330

GAP

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

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

C2×#1515165 #72337873