Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806671 |
| 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 | #42800 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1593 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)(11,16)(12,14)(13,15)(17,18)(19,29)(20,25)(21,37)(22,26)(23,28)(24,31)(27,32)(30,45)(33,41)(34,42)(35,40)(36,43)(38,44)(39,47)(46,48);(1,2,4,5)(3,6,7,8)(17,19,22,31)(18,24,26,29)(20,33,34,35)(21,38,32,23)(25,40,42,41)(27,44,37,28)(30,46,47,36)(39,48,45,43);(9,11,13,12)(10,14,15,16)(17,20,23,36,22,34,38,46)(18,25,28,43,26,42,44,48)(19,30,32,33,31,47,21,35)(24,39,37,40,29,45,27,41);(1,3,4,7)(2,6,5,8)(18,26)(20,34)(24,29)(27,37)(28,44)(30,47)(33,35)(36,46);(2,5)(6,8)(9,12,13,11)(10,14,15,16)(17,21,38,31,22,32,23,19)(18,27,44,29,26,37,28,24)(20,35,46,47,34,33,36,30)(25,41,48,45,42,40,43,39)Polycyclic presentation:
2^9|b^2=f,c^2=gh,d^2=fi,e^2=ghi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6744106);