Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7624150 |
| 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 | C23 | C22×D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #533 ◁ #2960 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1147 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)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18,23,28)(19,24,30,45)(20,33,36,67)(21,37,41,66)(22,42,39,27)(25,48,50,34)(26,51,53,40)(29,55,56,78)(31,61,59,77)(32,64,60,47)(35,68,43,52)(38,54,69,49)(44,70,71,57)(46,74,73,63)(58,79,65,75)(62,76,80,72);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19)(18,24)(20,29)(21,31)(22,32)(23,30)(25,44)(26,46)(27,47)(28,45)(33,55)(34,57)(35,58)(36,56)(37,61)(38,62)(39,60)(40,63)(41,59)(42,64)(43,65)(48,70)(49,72)(50,71)(51,74)(52,75)(53,73)(54,76)(66,77)(67,78)(68,79)(69,80);(17,20,23,36)(18,25,28,50)(19,29,30,56)(21,38,41,69)(22,43,39,35)(24,44,45,71)(26,52,53,68)(27,54,42,49)(31,62,59,80)(32,65,60,58)(33,66,67,37)(34,51,48,40)(46,75,73,79)(47,76,64,72)(55,77,78,61)(57,74,70,63);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15);(2,8)(6,9)(7,16)(14,15)(19,30)(24,45)(29,56)(31,59)(32,60)(44,71)(46,73)(47,64)(55,78)(57,70)(58,65)(61,77)(62,80)(63,74)(72,76)(75,79)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=i,d^2=f,g^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gh,h^a=hi,d^b=df,e^b=ei,g^c=gh,h^c=hi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7531595);