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C2×#499683 #72284334

order1024
nilpotency class4
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C28

Aliases

direct productC2×#499683

Subgroups

subgroupG/H
centerC2×C8#486
commutator, Frattini, cosocleC22×C8C25
socleC22#23815
prime radicalC2×#3150C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C8 ◁ C23×C8 ◁ C2×#1726 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

2949 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:

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

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#72283823, C2×#72283870, C2×#72283942
maximal normal subgroup ofC2×#72283823, C2×#72283870, C2×#72283942

GAP

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

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