C₂×#9166434 #72278583

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

Aliases

direct product

C₂×#9166434

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

At least 9936 conjugacy classes of subgroups; only the 29469 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 128

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

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

Conjugacy classes & character table

166 conjugacy classes — view the full character table.

Relationships

maximal quotient of

C₂×#72277923, C₂×#72278026, C₂×#72278043

GAP

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

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

C2×#9166434 #72278583