Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6370843 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C22×#117 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C2 | C22×#380 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×#568 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4425 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,7,12,8,13,6,27)(3,14,18,31,19,39,17,9)(4,20,22,33,23,10,28,34)(5,11,26,36,24,35,25,52)(15,40,42,48,43,37,47,29)(16,38,46,55,44,59,45,30)(21,49,51,56,50,32,53,57)(41,60,62,64,61,58,63,54);(1,3,6,17,8,19,7,18)(2,9,12,14,13,31,27,39)(4,15,22,42,23,43,28,47)(5,16,25,45,24,44,26,46)(10,29,33,37,20,48,34,40)(11,30,36,38,35,55,52,59)(21,41,51,62,50,61,53,63)(32,54,56,58,49,64,57,60);(1,4)(2,10)(3,15)(5,21)(6,22)(7,28)(8,23)(9,29)(11,32)(12,33)(13,20)(14,37)(16,41)(17,42)(18,47)(19,43)(24,50)(25,51)(26,53)(27,34)(30,54)(31,48)(35,49)(36,56)(38,58)(39,40)(44,61)(45,62)(46,63)(52,57)(55,64)(59,60);(1,5)(2,11)(3,16)(4,21)(6,25)(7,26)(8,24)(9,30)(10,32)(12,36)(13,35)(14,38)(15,41)(17,45)(18,46)(19,44)(20,49)(22,51)(23,50)(27,52)(28,53)(29,54)(31,55)(33,56)(34,57)(37,58)(39,59)(40,60)(42,62)(43,61)(47,63)(48,64);(5,24)(11,35)(16,44)(21,50)(25,26)(30,55)(32,49)(36,52)(38,59)(41,61)(45,46)(51,53)(54,64)(56,57)(58,60)(62,63)Polycyclic presentation:
2^9|a^2=h,b^2=f,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,h^c=hi,e^d=ei116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6278040);