Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7624824 |
| 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 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #533 ◁ #2960 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1163 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)(19,29)(20,34)(21,26)(22,42)(23,43)(24,31)(25,38)(27,36)(28,41)(30,53)(32,59)(33,46)(35,63)(37,60)(39,71)(40,72)(44,65)(45,64)(47,55)(48,69)(49,76)(50,68)(51,54)(52,58)(56,79)(57,73)(61,67)(62,70)(66,77)(74,78)(75,80);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19,21,31)(18,24,26,29)(20,35,37,30)(22,32,40,57)(23,33,41,58)(25,47,48,44)(27,45,50,51)(28,46,43,52)(34,53,60,63)(36,54,68,64)(38,65,69,55)(39,66,70,56)(42,73,72,59)(49,75,61,74)(62,77,71,79)(67,80,76,78);(17,20,36,67,23,39,40,69,21,37,68,76,41,70,22,38)(18,25,42,62,28,49,50,60,26,48,72,71,43,61,27,34)(19,30,54,78,33,56,57,65,31,35,64,80,58,66,32,55)(24,44,73,79,46,74,51,53,29,47,59,77,52,75,45,63);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(19,31)(24,29)(30,35)(32,57)(33,58)(44,47)(45,51)(46,52)(53,63)(54,64)(55,65)(56,66)(59,73)(74,75)(77,79)(78,80);(2,8)(6,9)(7,16)(14,15)(18,26)(19,31)(25,48)(27,50)(28,43)(30,35)(32,57)(33,58)(34,60)(42,72)(49,61)(54,64)(55,65)(56,66)(62,71)(78,80)Polycyclic presentation:
2^9|b^2=f,c^2=gh,d^2=fi,g^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gh,h^a=hi,c^b=ci,d^b=df,e^b=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7532138);