C₂×#775283 #72337923

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

Aliases

direct product

C₂×#775283

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

39707 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)(8,9)(11,12)(13,15)(14,16)(17,18)(19,20,23,25)(21,22,26,24);(7,8,10,9)(11,12,14,16)(13,15,18,17)(19,21,23,26)(20,24,25,22);(4,6)(7,9,10,8)(20,25)(21,26);(3,4)(5,6)(11,12)(13,17)(14,16)(15,18)(19,21)(20,24)(22,25)(23,26);(1,2);(11,13)(12,15)(14,18)(16,17)(19,22,23,24)(20,26,25,21)

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232339, C₂×#72232305, C₂×#72232350
maximal normal subgroup of	C₂×#72232339, C₂×#72232305, C₂×#72232350

GAP

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

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

C2×#775283 #72337923