Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806623 |
| 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 | #42882 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1425 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)(17,18,23,28)(19,24,30,37)(20,25,32,39)(21,26,34,41)(22,27,35,42)(29,36,44,47)(31,38,45,48)(33,40,43,46);(1,2)(3,6)(4,5)(7,8)(17,19,23,30)(18,24,28,37)(20,31,32,45)(21,22,34,35)(25,38,39,48)(26,27,41,42)(29,43,44,33)(36,46,47,40);(9,11,13,12)(10,14,15,16)(17,20,22,33,23,32,35,43)(18,25,27,40,28,39,42,46)(19,29,21,31,30,44,34,45)(24,36,26,38,37,47,41,48);(1,3,4,7)(2,6,5,8)(20,32)(25,39)(29,44)(31,45)(33,43)(36,47)(38,48)(40,46);(2,5)(6,8)(9,12,13,11)(10,14,15,16)(17,21,22,30,23,34,35,19)(18,26,27,37,28,41,42,24)(20,31,33,44,32,45,43,29)(25,38,40,47,39,48,46,36)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=gh,d^2=f,e^2=gh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6743511);