Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #392277 |
| 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 | |
3283 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)(5,22,24,34)(6,12)(7,11)(9,20)(10,23,32,25)(14,27,39,35)(15,41)(16,37,45,29)(19,48,49,54)(21,47)(26,33)(28,42)(30,52,46,50)(31,51)(36,44)(38,59)(40,56,62,58)(43,53)(55,63)(57,61)(60,64);(1,3)(2,8)(4,14)(5,15)(6,16)(7,17)(9,27)(10,28)(11,13)(12,29)(18,35)(19,38)(20,39)(21,40)(22,53)(23,42)(24,43)(25,44)(26,45)(30,55)(31,56)(32,36)(33,37)(34,41)(46,57)(47,58)(48,64)(49,60)(50,61)(51,62)(52,63)(54,59);(1,4)(2,9,11,18)(3,14)(5,19)(6,21)(7,20)(8,27,13,35)(10,30,32,46)(12,31,33,47)(15,38)(16,40)(17,39)(22,54,34,48)(23,50)(24,49)(25,52)(26,51)(28,55,36,57)(29,56,37,58)(41,64,53,59)(42,61)(43,60)(44,63)(45,62);(1,5,7,24)(2,10,11,32)(3,15,17,43)(4,19,20,49)(6,25,26,23)(8,28,13,36)(9,30,18,46)(12,34,33,22)(14,38,39,60)(16,44,45,42)(21,52,51,50)(27,55,35,57)(29,41,37,53)(31,48,47,54)(40,63,62,61)(56,64,58,59)Polycyclic presentation:
2^9|a^2=h,c^2=e,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,h^b=hi,f^c=fi,g^d=gi,h^d=hi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299474);