Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | #36684 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1467 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(3,7)(9,10)(11,13)(12,14)(15,16)(17,18,22,27,23,28,30,43)(19,24,21,35,31,39,37,26)(20,25,34,42,33,41,46,47)(29,38,36,40,45,48,32,44);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,24)(20,29)(21,22)(23,31)(25,38)(26,43)(27,35)(28,39)(30,37)(32,46)(33,45)(34,36)(40,42)(41,48)(44,47);(10,14)(11,15)(17,20,22,34,23,33,30,46)(18,25,27,42,28,41,43,47)(19,29,21,36,31,45,37,32)(24,38,35,40,39,48,26,44);(2,5)(6,8)(10,14)(13,16)(17,21)(18,26)(19,30)(20,32)(22,31)(23,37)(24,27)(25,40)(28,35)(29,34)(33,36)(38,47)(39,43)(41,44)(42,48)(45,46);(1,3,4,7)(2,6,5,8)(19,31)(20,33)(21,37)(24,39)(25,41)(26,35)(34,46)(42,47)Polycyclic presentation:
2^9|a^2=h,c^2=h,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dfi,e^b=ei,d^c=di,e^c=ei,e^d=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6319098);