Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32700 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2760 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14375 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1001 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,12,7,11,28,38)(3,13,17,39,16,8,26,31)(4,18,22,47,21,9,32,35)(5,23,25,40,24,10,27,37)(14,33,42,58,19,29,49,54)(15,36,44,56,43,30,45,55)(20,34,51,59,50,46,52,60)(41,57,61,64,48,53,62,63);(1,3)(2,8)(4,19)(5,15)(6,26)(7,16)(9,33)(10,36)(11,13)(12,39)(14,21)(17,28)(18,29)(20,41)(22,42)(23,30)(24,43)(25,45)(27,44)(31,38)(32,49)(34,57)(35,54)(37,55)(40,56)(46,53)(47,58)(48,50)(51,62)(52,61)(59,63)(60,64);(1,4)(2,9,11,18)(3,14,16,19)(5,20)(6,22)(7,21)(8,29)(10,34,23,46)(12,35,38,47)(13,33)(15,41,43,48)(17,42,26,49)(24,50)(25,51)(27,52)(28,32)(30,53)(31,54)(36,57)(37,59,40,60)(39,58)(44,61,45,62)(55,63)(56,64);(1,5)(2,10)(3,15)(4,20)(6,27)(7,24)(8,30)(9,34)(11,23)(12,40)(13,36)(14,41)(16,43)(17,45)(18,46)(19,48)(21,50)(22,52)(25,28)(26,44)(29,53)(31,56)(32,51)(33,57)(35,60)(37,38)(39,55)(42,62)(47,59)(49,61)(54,64)(58,63)Polycyclic presentation:
2^9|a^2=h,c^2=ef,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,f^b=fi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,g^d=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 291458);