Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #364604 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13967 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2330 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2)(3,14)(4,20,24,35)(5,25)(6,28)(7,12)(8,13)(9,17)(10,23,36,30)(11,26)(15,31,21,49)(16,32)(18,33)(19,39)(22,51,54,34)(27,37)(29,38)(40,43,56,47)(41,44)(42,61,50,60)(45,58)(46,57)(48,53,59,52)(55,63,64,62);(1,3)(2,9)(4,21,24,15)(5,16)(6,18)(7,17)(8,19)(10,31,36,49)(11,32)(12,39)(13,33)(14,28)(20,40,35,56)(22,50,54,42)(23,43,30,47)(25,41)(26,46)(27,45)(29,44)(34,55,51,64)(37,57)(38,58)(48,60,59,61)(52,62,53,63);(1,4)(2,10)(3,15)(5,22)(6,23)(7,30)(8,24)(9,31)(11,34)(12,35)(13,36)(14,40)(16,42)(17,43)(18,47)(19,21)(20,28)(25,48)(26,52)(27,54)(29,53)(32,55)(33,49)(37,59)(38,51)(39,56)(41,60)(44,62)(45,50)(46,63)(57,61)(58,64);(1,5,8,27)(2,11,13,38)(3,16,19,45)(4,22,24,54)(6,29,7,26)(9,32,33,58)(10,34,36,51)(12,37,28,25)(14,41,39,57)(15,42,21,50)(17,46,18,44)(20,48,35,59)(23,53,30,52)(31,55,49,64)(40,60,56,61)(43,63,47,62)Polycyclic presentation:
2^9|a^2=h,b^2=h,d^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,f^c=fi,h^c=hi,e^d=ei77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 271801);