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 | #61477 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #529 ◁ #2966 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1175 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)(17,18,22,28)(19,31,34,25)(20,37,39,71)(21,30,24,27)(23,43,42,54)(26,51,53,40)(29,41,55,45)(32,60,62,70)(33,48,36,50)(35,66,65,74)(38,72,46,56)(44,57,73,52)(47,63,58,78)(49,68,59,64)(61,76,69,79)(67,75,80,77);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,15)(13,16)(17,19,22,34)(18,25,28,31)(20,32,39,62)(21,33,24,36)(23,35,42,65)(26,47,53,58)(27,48,30,50)(29,49,55,59)(37,70,71,60)(38,61,46,69)(40,63,51,78)(41,64,45,68)(43,74,54,66)(44,67,73,80)(52,75,57,77)(56,76,72,79);(17,20)(18,26)(19,32)(21,38)(22,39)(23,44)(24,46)(25,47)(27,52)(28,53)(29,56)(30,57)(31,58)(33,61)(34,62)(35,67)(36,69)(37,43)(40,45)(41,51)(42,73)(48,75)(49,76)(50,77)(54,71)(55,72)(59,79)(60,66)(63,68)(64,78)(65,80)(70,74);(3,10)(6,9)(11,16)(14,15)(17,21,22,24)(18,27,28,30)(19,33,34,36)(20,38,39,46)(23,41,42,45)(25,48,31,50)(26,52,53,57)(29,54,55,43)(32,61,62,69)(35,64,65,68)(37,56,71,72)(40,44,51,73)(47,75,58,77)(49,66,59,74)(60,79,70,76)(63,67,78,80);(1,4,5,13)(2,7,8,12)(3,11,10,16)(6,14,9,15)(18,28)(19,34)(26,53)(27,30)(29,55)(32,62)(33,36)(35,65)(37,71)(43,54)(52,57)(56,72)(61,69)(63,78)(64,68)(67,80)Polycyclic presentation:
2^9|a^2=f,b^2=i,d^2=i,e^2=fi,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, 7531992);