C₂×#8807309 #72278490

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁸

Aliases

direct product

C₂×#8807309

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂⁶
commutator, Frattini, cosocle	C₂³	C₂⁷
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂³ ▷ C₁

55093 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 7 generators on 26 points:

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

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72278677
maximal normal subgroup of	#72278677

GAP

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

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

C2×#8807309 #72278490