Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6343554 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×C22⋊C4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | #62828 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2712 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(2,8,11,27)(3,14)(4,18)(5,21)(9,29,32,45)(10,34,36,53)(12,28,13,22)(15,38)(16,43)(19,46)(20,49)(23,50)(25,30,52,33)(26,35,54,37)(31,58,59,51)(39,41)(44,63)(47,48)(56,60,61,57)(62,64);(1,2)(3,9)(4,10)(5,22)(6,12)(7,13)(8,23)(11,24)(14,33)(15,30)(16,31)(17,45)(18,37)(19,35)(20,34)(21,27)(25,42)(26,47)(28,50)(29,55)(32,38)(36,46)(39,60)(40,52)(41,57)(43,59)(44,61)(48,54)(49,53)(51,63)(56,64)(58,62);(3,15)(4,19)(9,30)(10,35)(14,38)(16,43)(17,42)(18,46)(20,48)(25,45)(26,53)(29,52)(31,59)(32,33)(34,54)(36,37)(39,41)(40,55)(44,64)(47,49)(51,58)(56,61)(57,60)(62,63);(1,3,6,14)(2,9,12,33)(4,16,19,39)(5,17,21,40)(7,15,24,38)(8,25,28,29)(10,31,35,60)(11,32,13,30)(18,41,46,43)(20,44,48,62)(22,45,27,52)(23,42,50,55)(26,51,53,56)(34,61,54,58)(36,59,37,57)(47,63,49,64);(1,4)(2,10)(3,16)(5,20)(6,19)(7,18)(8,26)(9,31)(11,36)(12,35)(13,37)(14,39)(15,41)(17,44)(21,48)(22,34)(23,47)(24,46)(25,51)(27,54)(28,53)(29,56)(30,57)(32,59)(33,60)(38,43)(40,62)(42,63)(45,61)(49,50)(52,58)(55,64)Polycyclic presentation:
2^9|a^2=g,d^2=h,f^2=i,b^a=bf,d^a=dh,e^a=ei,f^a=fh,f^b=fi,g^b=ghi,d^c=di,e^c=eh92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6250906);