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 | |
4465 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)(3,13)(4,18,21,9)(5,10)(6,24)(7,12)(8,16)(11,25)(14,36,39,33)(15,34)(17,35)(19,44,46,29)(20,30,48,45)(22,31)(23,50)(26,38,43,59)(27,41)(28,42)(32,49)(37,56,57,54)(40,55)(47,53,63,62)(51,58,61,64)(52,60);(1,3,6,16,7,17,25,42)(2,8,11,13,12,28,24,35)(4,14,20,38,21,39,48,59)(5,15,23,41,22,40,49,60)(9,26,30,33,18,43,45,36)(10,27,32,34,31,52,50,55)(19,37,47,58,46,57,63,64)(29,51,53,54,44,61,62,56);(1,4)(2,9)(3,14)(5,19)(6,20)(7,21)(8,26)(10,29)(11,30)(12,18)(13,33)(15,37)(16,38)(17,39)(22,46)(23,47)(24,45)(25,48)(27,51)(28,43)(31,44)(32,53)(34,54)(35,36)(40,57)(41,58)(42,59)(49,63)(50,62)(52,61)(55,56)(60,64);(1,5)(2,10)(3,15)(4,19)(6,23)(7,22)(8,27)(9,29)(11,32)(12,31)(13,34)(14,37)(16,41)(17,40)(18,44)(20,47)(21,46)(24,50)(25,49)(26,51)(28,52)(30,53)(33,54)(35,55)(36,56)(38,58)(39,57)(42,60)(43,61)(45,62)(48,63)(59,64);(5,22)(10,31)(15,40)(19,46)(23,49)(27,52)(29,44)(32,50)(34,55)(37,57)(41,60)(47,63)(51,61)(53,62)(54,56)(58,64)Polycyclic presentation:
2^9|a^2=h,b^2=f,f^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, 6278039);