C₂×#10086476 #72329751

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

Aliases

direct product

C₂×#10086476

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 9746 conjugacy classes of subgroups; only the 29477 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)(6,10)(7,9)(11,12)(13,16)(14,18)(15,17);(3,5)(4,8)(6,9)(7,10)(11,13)(12,16)(14,17)(15,18)(19,20,22,23)(21,25,26,24);(4,9)(5,10)(11,12)(13,14)(15,17)(16,18)(20,23)(21,26);(4,9)(6,8)(19,20,22,23)(21,25,26,24);(1,2);(3,4)(5,6)(7,9)(8,10)(11,12)(13,16)(14,18)(15,17)(19,20)(21,25)(22,23)(24,26);(3,6)(4,5)(7,8)(9,10)(11,14,15,16)(12,13,17,18)(19,21)(20,24)(22,26)(23,25)

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232280
maximal normal subgroup of	C₂×#72232280
maximal quotient of	C₂×#72323415

GAP

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

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

C2×#10086476 #72329751