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 | #2535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1437 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,6)(3,7,4,8)(9,10)(11,15)(12,13)(14,16)(17,18)(19,24)(20,31)(21,35)(22,37)(23,28)(25,34)(26,30)(27,38)(29,41)(32,39)(33,47)(36,40)(42,48)(43,44)(45,46);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,23,30)(18,24,28,26)(20,32,33,48)(21,22,36,38)(25,41,43,46)(27,35,37,40)(29,44,45,34)(31,39,47,42);(10,13)(14,15)(17,20,22,34,23,33,38,44)(18,25,27,31,28,43,37,47)(19,29,21,32,30,45,36,48)(24,39,40,41,26,42,35,46);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,14,13,15)(17,21,22,30,23,36,38,19)(18,26,27,35,28,24,37,40)(20,32,34,45,33,48,44,29)(25,42,31,46,43,39,47,41);(1,4,5,3)(2,7,6,8)(20,33)(25,43)(29,45)(31,47)(32,48)(34,44)(39,42)(41,46)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=h,d^2=fg,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7505513);