Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7624170 |
| 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 | C2×#135 | C23 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #533 ◁ #2960 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1200 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)(17,18,22,28)(19,31,34,25)(20,36,38,70)(21,30,24,27)(23,43,42,54)(26,50,52,39)(29,41,55,45)(32,60,62,69)(33,48,40,53)(35,67,66,74)(37,71,46,56)(44,57,73,51)(47,63,58,78)(49,65,59,72)(61,79,64,77)(68,76,80,75);(1,3,6,13)(2,7,10,11)(4,12,5,14)(8,15,9,16)(17,19,22,34)(18,25,28,31)(20,32,38,62)(21,40,24,33)(23,35,42,66)(26,47,52,58)(27,53,30,48)(29,49,55,59)(36,69,70,60)(37,64,46,61)(39,63,50,78)(41,72,45,65)(43,74,54,67)(44,68,73,80)(51,76,57,75)(56,77,71,79);(1,4,6,5)(2,8,10,9)(3,12,13,14)(7,15,11,16)(17,20)(18,26)(19,32)(21,37)(22,38)(23,44)(24,46)(25,47)(27,51)(28,52)(29,56)(30,57)(31,58)(33,61)(34,62)(35,68)(36,43)(39,45)(40,64)(41,50)(42,73)(48,75)(49,77)(53,76)(54,70)(55,71)(59,79)(60,67)(63,65)(66,80)(69,74)(72,78);(1,5,6,4)(2,9,10,8)(3,12,13,14)(7,15,11,16);(2,10)(7,11)(8,9)(15,16)(17,21,22,24)(18,27,28,30)(19,33,34,40)(20,37,38,46)(23,41,42,45)(25,48,31,53)(26,51,52,57)(29,54,55,43)(32,61,62,64)(35,65,66,72)(36,56,70,71)(39,44,50,73)(47,75,58,76)(49,74,59,67)(60,77,69,79)(63,80,78,68)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=fi,d^2=fi,e^2=i,g^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gh,h^a=hi,d^b=dfi,e^b=ei,g^c=gh,h^c=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7531679);