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 | #42664 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1595 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)(19,24)(20,25)(21,34)(22,27)(23,28)(26,35)(29,37)(30,38)(31,39)(32,40)(33,42)(36,43)(41,46)(44,47)(45,48);(1,2,4,5)(3,6,7,8)(17,19,23,30)(18,24,28,38)(20,31,32,46)(21,22,35,36)(25,39,40,41)(26,43,34,27)(29,44,45,33)(37,47,48,42);(9,11,13,12)(10,14,15,16)(17,20,22,33,23,32,36,44)(18,25,27,42,28,40,43,47)(19,29,21,31,30,45,35,46)(24,37,34,39,38,48,26,41);(1,3,4,7)(2,6,5,8)(20,32)(25,40)(29,45)(31,46)(33,44)(37,48)(39,41)(42,47);(2,5)(6,8)(9,12,13,11)(10,14,15,16)(17,21,22,30,23,35,36,19)(18,26,27,24,28,34,43,38)(20,31,33,45,32,46,44,29)(25,41,42,37,40,39,47,48)Polycyclic presentation:
2^9|b^2=fi,c^2=gh,d^2=f,e^2=gh,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, 6744101);