C₂×#689728 #72337992

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

Aliases

direct product

C₂×#689728

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

42519 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:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232291, C₂×#72232296, C₂×#72232298
maximal normal subgroup of	C₂×#72232291, C₂×#72232296, C₂×#72232298

GAP

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

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

C2×#689728 #72337992