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 | #62827 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2710 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)(5,22)(9,29,32,45)(10,34,36,53)(12,28,13,23)(15,38)(16,43)(19,46)(21,50)(24,49)(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)(5,23)(6,12)(7,13)(8,24)(11,20)(14,33)(15,30)(16,31)(17,45)(18,37)(19,35)(21,34)(22,27)(25,42)(26,47)(28,49)(29,55)(32,38)(36,46)(39,60)(40,52)(41,57)(43,59)(44,61)(48,54)(50,53)(51,63)(56,64)(58,62);(3,15)(4,19)(9,30)(10,35)(14,38)(16,43)(17,42)(18,46)(21,48)(25,45)(26,53)(29,52)(31,59)(32,33)(34,54)(36,37)(39,41)(40,55)(44,64)(47,50)(51,58)(56,61)(57,60)(62,63);(1,3,6,14)(2,9,12,33)(4,16,19,39)(5,17,22,40)(7,15,20,38)(8,25,28,29)(10,31,35,60)(11,32,13,30)(18,41,46,43)(21,44,48,62)(23,45,27,52)(24,42,49,55)(26,51,53,56)(34,61,54,58)(36,59,37,57)(47,63,50,64);(1,4,20,46)(2,10,11,36)(3,16,38,43)(5,21,49,50)(6,19,7,18)(8,26,27,54)(9,31,32,59)(12,35,13,37)(14,39,15,41)(17,44,55,64)(22,48,24,47)(23,34,28,53)(25,51,52,58)(29,56,45,61)(30,57,33,60)(40,62,42,63)Polycyclic presentation:
2^9|a^2=g,d^2=h,e^2=hi,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, 6250926);