Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| 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 | |
1469 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)(17,18,22,27,23,28,30,37)(19,24,21,26,31,38,35,42)(20,25,34,41,33,40,43,46)(29,36,32,39,44,47,45,48);(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)(25,36)(26,27)(28,38)(30,35)(32,34)(33,44)(37,42)(39,41)(40,47)(43,45)(46,48);(10,14)(11,15)(17,20,22,34,23,33,30,43)(18,25,27,41,28,40,37,46)(19,29,21,32,31,44,35,45)(24,36,26,39,38,47,42,48);(2,5)(6,8)(10,14)(13,16)(17,21)(18,26)(19,30)(20,32)(22,31)(23,35)(24,37)(25,39)(27,38)(28,42)(29,43)(33,45)(34,44)(36,46)(40,48)(41,47);(1,3,4,7)(2,6,5,8)(19,31)(20,33)(21,35)(24,38)(25,40)(26,42)(34,43)(41,46)Polycyclic presentation:
2^9|a^2=h,c^2=h,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dfi,e^b=ei,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, 6319088);