Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7624184 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| prime radical | #33787 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #533 ◁ #2960 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1206 conjugacy classes of subgroups.
| minimal degree | μ(G) = 80 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 80 points:
(1,2)(3,11)(4,8)(5,9)(6,10)(7,13)(12,16)(14,15)(17,18)(19,31)(20,36)(21,30)(22,27)(23,43)(24,28)(25,34)(26,39)(29,45)(32,59)(33,48)(35,66)(37,70)(38,69)(40,53)(41,52)(42,54)(44,56)(46,55)(47,75)(49,72)(50,71)(51,73)(57,62)(58,64)(60,79)(61,68)(63,78)(65,74)(67,77)(76,80);(1,3,6,13)(2,7,10,11)(4,12,5,14)(8,15,9,16)(17,19)(18,25)(20,32)(21,40)(22,34)(23,35)(24,33)(26,47)(27,31)(28,53)(29,49)(30,48)(36,68)(37,63)(38,61)(39,62)(41,72)(42,65)(43,74)(44,67)(45,64)(46,60)(50,57)(51,77)(52,58)(54,66)(55,78)(56,76)(59,69)(70,79)(71,75)(73,80);(1,4,6,5)(2,8,10,9)(3,12,13,14)(7,15,11,16)(17,20,22,38)(18,26,27,50)(19,32,34,61)(21,37,24,46)(23,44,42,73)(25,47,31,57)(28,51,30,56)(29,55,52,70)(33,60,40,63)(35,67,65,80)(36,54,69,43)(39,41,71,45)(48,76,53,77)(49,78,58,79)(59,74,68,66)(62,72,75,64);(1,5,6,4)(2,9,10,8)(3,12,13,14)(7,15,11,16)(18,27)(25,31)(26,50)(28,30)(29,52)(36,69)(43,54)(47,57)(48,53)(49,58)(51,56)(55,70)(59,68)(66,74)(76,77)(78,79);(2,10)(7,11)(8,9)(15,16)(17,21,22,24)(18,28,27,30)(19,33,34,40)(20,37,38,46)(23,41,42,45)(25,48,31,53)(26,51,50,56)(29,54,52,43)(32,60,61,63)(35,64,65,72)(36,55,69,70)(39,44,71,73)(47,76,57,77)(49,74,58,66)(59,78,68,79)(62,80,75,67)Polycyclic presentation:
2^9|b^2=fi,c^2=f,d^2=fi,e^2=i,g^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,g^a=gh,h^a=hi,d^b=dfi,e^b=ei,g^c=gh,h^c=hi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7531689);