Finite Group Databasebeta

← Prev (#72308143) Next (#72308145) →

C2×#500941 #72308144

order1024
nilpotency class4
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C29

Aliases

direct productC2×#500941

Subgroups

subgroupG/H
center, socleC22#24338
commutator, Frattini, cosocleC22×C8C25
prime radicalC22×#135C23
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#2961 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

4584 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 66
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 5 generators on 66 points:

(1,2);(3,4)(5,17,22,45)(6,24)(7,29)(8,33)(9,36)(10,11)(12,21,37,31)(13,25)(14,19)(15,30)(16,35)(18,40,50,47)(20,48,53,62)(23,38,34,41)(26,57)(27,46)(28,44)(32,43)(39,55,56,52)(42,51)(49,63,66,65)(54,61,60,64)(58,59);(4,11)(5,18,22,50)(6,25)(7,30)(8,19)(12,38,37,41)(16,36)(17,47,45,40)(20,49,53,66)(21,23,31,34)(28,59)(39,61,56,64)(42,46)(43,57)(48,65,62,63)(52,60,55,54);(3,5,19,23,35,50,7,31,10,22,8,34,9,18,30,21)(4,12,29,40,36,38,14,45,11,37,15,47,16,41,33,17)(6,20,51,54,59,66,32,52,25,53,27,60,28,49,26,55)(13,39,57,63,58,61,46,62,24,56,43,65,44,64,42,48);(3,6,10,25)(4,13,11,24)(5,20)(7,32,30,26)(8,27,19,51)(9,28,35,59)(12,39)(14,46,33,42)(15,43,29,57)(16,44,36,58)(17,48)(18,49)(21,55)(22,53)(23,54)(31,52)(34,60)(37,56)(38,61)(40,63)(41,64)(45,62)(47,65)(50,66)

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#72307107, C2×#72307114, C2×#72307115
maximal normal subgroup ofC2×#72307107, C2×#72307114, C2×#72307115

GAP

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

gap> G := Group((1,2), (3,4)(5,17,22,45)(6,24)(7,29)(8,33)(9,36)(10,11)(12,21,37,31)(13,25)(14,19)(15,30)(16,35)(18,40,50,47)(20,48,53,62)(23,38,34,41)(26,57)(27,46)(28,44)(32,43)(39,55,56,52)(42,51)(49,63,66,65)(54,61,60,64)(58,59), (4,11)(5,18,22,50)(6,25)(7,30)(8,19)(12,38,37,41)(16,36)(17,47,45,40)(20,49,53,66)(21,23,31,34)(28,59)(39,61,56,64)(42,46)(43,57)(48,65,62,63)(52,60,55,54), (3,5,19,23,35,50,7,31,10,22,8,34,9,18,30,21)(4,12,29,40,36,38,14,45,11,37,15,47,16,41,33,17)(6,20,51,54,59,66,32,52,25,53,27,60,28,49,26,55)(13,39,57,63,58,61,46,62,24,56,43,65,44,64,42,48), (3,6,10,25)(4,13,11,24)(5,20)(7,32,30,26)(8,27,19,51)(9,28,35,59)(12,39)(14,46,33,42)(15,43,29,57)(16,44,36,58)(17,48)(18,49)(21,55)(22,53)(23,54)(31,52)(34,60)(37,56)(38,61)(40,63)(41,64)(45,62)(47,65)(50,66));