Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519008 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×D8 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2402 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1348 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,7)(4,8)(5,6)(9,10,12,14)(11,15,16,13)(17,18)(19,24)(20,30)(21,23)(22,26)(25,33)(27,40)(28,35)(29,38)(31,39)(32,37)(34,36)(41,45)(42,47)(43,46)(44,48);(2,6)(7,8)(11,16)(13,15)(18,23)(20,31)(22,34)(24,35)(25,39)(27,42)(29,43)(30,41)(32,44)(33,45)(37,47)(40,48);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,13,14,15)(17,19)(18,24)(20,32)(21,28)(22,29)(23,35)(25,40)(26,38)(27,33)(30,37)(31,44)(34,43)(36,46)(39,48)(41,47)(42,45);(10,14)(13,15)(17,20,22,31,21,33,34,45)(18,25,26,41,23,30,36,39)(19,27,29,42,28,32,43,44)(24,37,38,48,35,40,46,47);(1,4,5,3)(2,7,6,8)(18,23)(19,28)(20,33)(26,36)(29,43)(31,45)(37,40)(47,48)Polycyclic presentation:
2^9|a^2=g,c^2=fgi,d^2=h,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7522799);