C₂×#4941671 #72324399

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁵

Aliases

direct product

C₂×#4941671

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

59169 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:

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72219264, C₂×#72232060, C₂×#72232047, …
maximal normal subgroup of	C₂×#72219264, C₂×#72232060, C₂×#72232047, …

GAP

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

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

C2×#4941671 #72324399