Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #449383 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15941 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
805 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)(4,19)(5,25)(6,30)(7,13)(8,12)(9,17)(10,23)(11,29)(15,36)(16,39)(18,35)(20,47)(21,45)(22,38)(24,46)(26,33)(27,40)(28,55)(31,34)(32,49)(37,53)(41,43)(42,61)(44,54)(48,57)(50,59)(51,60)(52,63)(56,58)(62,64);(1,3,8,31,7,18,6,17)(2,9,30,35,13,34,12,14)(4,20,24,15,23,49,22,43)(5,26,27,44,29,56,28,16)(10,36,46,47,19,41,38,32)(11,39,40,58,25,54,55,33)(21,50,51,64,53,42,52,48)(37,57,60,61,45,62,63,59);(1,4,6,22,7,23,8,24)(2,10,30,46,13,19,12,38)(3,15,31,49,18,43,17,20)(5,21,27,51,29,53,28,52)(9,32,14,41,34,47,35,36)(11,37,55,63,25,45,40,60)(16,42,56,64,44,50,26,48)(33,57,39,61,58,62,54,59);(1,5)(2,11,13,25)(3,16,18,44)(4,21)(6,27)(7,29)(8,28)(9,33)(10,37,19,45)(12,40,30,55)(14,39)(15,42,43,50)(17,26,31,56)(20,48,49,64)(22,51)(23,53)(24,52)(32,57)(34,58)(35,54)(36,59)(38,60,46,63)(41,61)(47,62)Polycyclic presentation:
2^9|b^2=ef,c^2=gi,d^2=eg,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,g^b=gi,h^b=hi,h^c=hi,f^d=fi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 356580);