C₂×#773116 #72299178

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴

Aliases

direct product

C₂×#773116

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

28863 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 6 generators on 26 points:

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

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72297589, C₂×#72282047
maximal normal subgroup of	C₂×#72297589, C₂×#72282047

GAP

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

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

C2×#773116 #72299178