C₂×#6357800 #72224793

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²×C₈

Aliases

direct product

C₂×#6357800

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×#127
commutator, Frattini, cosocle	C₄²	C₂⁶
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

18413 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:

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

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72224795

GAP

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

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

C2×#6357800 #72224793