C₂×#8433751 #72294792

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

Aliases

direct product

C₂×#8433751

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 10102 conjugacy classes of subgroups; only the 29573 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)(6,7)(9,10)(11,12,15,17)(13,16,18,14)(19,20)(21,22)(23,26)(24,25);(5,9)(8,10)(11,12)(13,14)(15,17)(16,18)(19,21,23,25)(20,24,26,22);(4,7)(5,9)(13,18)(14,16)(19,22,23,24)(20,25,26,21);(3,4)(5,8)(6,7)(9,10)(11,13,15,18)(12,16,17,14)(19,22)(20,21)(23,24)(25,26);(3,4)(5,8)(6,7)(9,10)(11,13)(12,14)(15,18)(16,17)(19,22)(20,21)(23,24)(25,26);(3,5)(4,8)(6,9)(7,10)(11,14)(12,13)(15,16)(17,18)(19,21)(20,22)(23,25)(24,26);(1,2)

Conjugacy classes & character table

178 conjugacy classes — view the full character table.

Relationships

maximal quotient of

C₂×#72293915, C₂×#72293917, C₂×#72293925

GAP

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

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

C2×#8433751 #72294792