Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #392237 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14681 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3289 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)(3,13,17,8)(4,18,21,48)(5,10)(6,12)(7,11)(9,20,30,53)(14,38)(15,36,43,27)(16,37,45,28)(19,49,51,47)(22,32)(23,31)(24,34)(25,33)(26,40)(29,50,57,52)(35,58)(39,61)(41,46)(42,56,44,55)(54,62)(59,64)(60,63);(1,3)(2,8)(4,14)(5,15)(6,16)(7,17)(9,26)(10,27)(11,13)(12,28)(18,46)(19,39)(20,40)(21,41)(22,42)(23,43)(24,44)(25,45)(29,54)(30,35)(31,36)(32,55)(33,37)(34,56)(38,48)(47,61)(49,60)(50,62)(51,63)(52,64)(53,58)(57,59);(1,4)(2,9,11,30)(3,14)(5,19)(6,20)(7,21)(8,26,13,35)(10,29,31,57)(12,18,33,48)(15,39)(16,40)(17,41)(22,50)(23,51)(24,52)(25,53)(27,54,36,59)(28,46,37,38)(32,49,34,47)(42,62)(43,63)(44,64)(45,58)(55,60,56,61);(1,5,7,23)(2,10,11,31)(3,15,17,43)(4,19,21,51)(6,24,25,22)(8,27,13,36)(9,29,30,57)(12,34,33,32)(14,39,41,63)(16,44,45,42)(18,47,48,49)(20,52,53,50)(26,54,35,59)(28,56,37,55)(38,60,46,61)(40,64,58,62)Polycyclic presentation:
2^9|a^2=h,c^2=e,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,h^b=hi,g^c=gi,h^c=hi,f^d=fi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299434);