Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5956507 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1196 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 64 points:
(3,9)(4,11)(6,12)(7,14)(19,25)(23,28)(26,31)(30,32)(33,34,37,40)(35,38,41,42)(36,39,44,46)(43,45,47,48)(49,50)(51,54)(52,60)(53,56)(55,61)(57,59)(58,63)(62,64);(3,9)(6,12)(10,15)(13,16)(18,22)(19,25)(24,29)(26,31)(35,41)(38,42)(43,47)(45,48)(49,50,53,56)(51,54,59,57)(52,60,61,55)(58,63,64,62);(1,2,5,8)(3,6,9,12)(4,7,11,14)(10,13,15,16)(17,18,21,22)(19,23,25,28)(20,24,27,29)(26,30,31,32)(33,34)(35,42)(36,39)(37,40)(38,41)(43,48)(44,46)(45,47)(49,50)(51,57)(52,55)(53,56)(54,59)(58,63)(60,61)(62,64);(1,3,5,9)(2,6,8,12)(4,10,11,15)(7,13,14,16)(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32)(33,35,37,41)(34,38,40,42)(36,43,44,47)(39,45,46,48)(49,51,53,59)(50,54,56,57)(52,58,61,64)(55,62,60,63);(1,4)(2,7)(3,10)(5,11)(6,13)(8,14)(9,15)(12,16)(17,20,21,27)(18,24,22,29)(19,26,25,31)(23,30,28,32)(33,36,37,44)(34,39,40,46)(35,43,41,47)(38,45,42,48)(49,52)(50,55)(51,58)(53,61)(54,62)(56,60)(57,63)(59,64)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=i,d^2=ghi,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=ef,d^c=dfg62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6018097);