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 | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1399 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,8)(3,7,4,6)(9,10)(11,13)(12,14)(15,16)(17,18,21,23)(19,24,28,35)(20,30,34,25)(22,26,33,36)(27,40,32,37)(29,38,43,46)(31,39,45,41)(42,47,44,48);(1,3,5,4)(2,6,8,7)(9,11)(10,13)(12,15)(14,16)(18,23)(20,31)(22,33)(24,35)(25,39)(27,42)(29,43)(30,41)(32,44)(34,45)(37,47)(40,48);(1,3,5,4)(2,7,8,6)(10,14)(11,15)(17,19,21,28)(18,24,23,35)(20,32,34,27)(22,29,33,43)(25,40,30,37)(26,38,36,46)(31,44,45,42)(39,48,41,47);(10,14)(13,16)(17,20,33,45,21,34,22,31)(18,25,36,39,23,30,26,41)(19,27,43,44,28,32,29,42)(24,37,46,47,35,40,38,48);(1,4,5,3)(2,7,8,6)(18,23)(19,28)(20,34)(26,36)(29,43)(31,45)(37,40)(47,48)Polycyclic presentation:
2^9|a^2=f,b^2=fi,c^2=f,d^2=hi,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, 7520527);