C₂×#647814 #72337943

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

Aliases

direct product

C₂×#647814

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂×#14715	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

49757 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:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232322, C₂×#72232339, C₂×#72232295
maximal normal subgroup of	C₂×#72232322, C₂×#72232339, C₂×#72232295

GAP

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

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

C2×#647814 #72337943