Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30496 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2669 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14361 |
| prime radical | #60469 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1344 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)(3,15)(4,21,26,37)(5,27,31,11)(6,13)(7,12)(8,14)(9,18)(10,25,38,24)(16,32,45,49)(17,39,47,56)(19,34)(20,41)(22,35,48,43)(23,50)(28,33,46,59)(29,40,30,42)(36,53)(44,63)(51,62)(52,58)(54,61)(55,57)(60,64);(1,3)(2,9)(4,22)(5,28)(6,19)(7,18)(8,20)(10,35)(11,39)(12,41)(13,15)(14,34)(16,24)(17,29)(21,49)(23,52)(25,45)(26,48)(27,56)(30,47)(31,46)(32,37)(33,40)(36,58)(38,43)(42,59)(44,54)(50,60)(51,53)(55,64)(57,63)(61,62);(1,4)(2,10)(3,16)(5,23)(6,24)(7,25)(8,26)(9,32)(11,36)(12,21)(13,37)(14,38)(15,43)(17,44)(18,22)(19,48)(20,45)(27,57)(28,51)(29,53)(30,55)(31,54)(33,58)(34,49)(35,41)(39,62)(40,50)(42,61)(46,64)(47,52)(56,60)(59,63);(1,5,6,29,8,31,7,30)(2,11,12,40,14,27,13,42)(3,17,18,28,20,47,19,46)(4,23,24,53,26,54,25,55)(9,33,15,39,34,59,41,56)(10,36,21,50,38,57,37,61)(16,44,22,51,45,52,48,64)(32,58,43,62,49,63,35,60)Polycyclic presentation:
2^9|a^2=h,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,e^b=ei,g^b=gi,f^c=fi,g^c=gi,h^c=hi,f^d=fi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290737);