C₂×#10086698 #72331998

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

Aliases

direct product

C₂×#10086698

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 9748 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:

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

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232288
maximal normal subgroup of	C₂×#72232288
maximal quotient of	C₂×#72323388

GAP

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

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

C2×#10086698 #72331998