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 | #121 | D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #529 ◁ #2966 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1172 conjugacy classes of subgroups.
| minimal degree | μ(G) = 80 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 80 points:
(1,2,5,8)(3,9,10,6)(4,12,13,7)(11,14,16,15)(17,18)(19,29)(20,34)(21,26)(22,40)(23,43)(24,31)(25,36)(27,42)(28,39)(30,56)(32,62)(33,47)(35,68)(37,71)(38,67)(41,52)(44,51)(45,74)(46,76)(48,69)(49,72)(50,70)(53,58)(54,64)(55,61)(57,66)(59,77)(60,73)(63,75)(65,79)(78,80);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,15)(13,16)(17,19)(18,24)(20,30)(21,31)(22,32)(23,33)(25,45)(26,29)(27,46)(28,47)(34,66)(35,57)(36,58)(37,59)(38,60)(39,61)(40,73)(41,63)(42,64)(43,55)(44,65)(48,53)(49,75)(50,54)(51,77)(52,78)(56,68)(62,67)(69,74)(70,76)(71,79)(72,80);(17,20,21,35)(18,25,26,48)(19,30,31,57)(22,41,38,72)(23,44,39,37)(24,45,29,53)(27,51,50,71)(28,52,43,49)(32,63,60,80)(33,65,61,59)(34,67,68,40)(36,70,69,42)(46,77,54,79)(47,78,55,75)(56,73,66,62)(58,76,74,64);(3,10)(6,9)(11,16)(14,15)(19,31)(24,29)(30,57)(32,60)(33,61)(45,53)(46,54)(47,55)(56,66)(58,74)(59,65)(62,73)(63,80)(64,76)(75,78)(77,79);(1,4,5,13)(2,7,8,12)(3,11,10,16)(6,14,9,15)(18,26)(19,31)(25,48)(27,50)(28,43)(30,57)(32,60)(33,61)(34,68)(40,67)(49,52)(51,71)(58,74)(59,65)(63,80)(64,76)Polycyclic presentation:
2^9|a^2=fi,c^2=i,e^2=fi,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, 7531923);