Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806647 |
| 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 | #42926 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1684 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)(3,7)(4,6)(5,8)(9,10)(11,14)(12,13)(15,16)(17,18,24,30)(19,25,34,43)(20,26,35,44)(21,27,23,29)(22,28,38,47)(31,40,37,46)(32,41,39,48)(33,42,36,45);(1,3)(2,5)(4,8)(6,7)(17,19,24,34)(18,25,30,43)(20,22,35,38)(21,32,23,39)(26,28,44,47)(27,41,29,48)(31,36,37,33)(40,45,46,42);(10,13)(14,16)(17,20,21,36,24,35,23,33)(18,26,27,45,30,44,29,42)(19,31,39,38,34,37,32,22)(25,40,48,47,43,46,41,28);(2,6)(5,7)(17,21,24,23)(18,27,30,29)(19,32,34,39)(20,33,35,36)(22,31,38,37)(25,41,43,48)(26,42,44,45)(28,40,47,46);(3,8)(5,7)(9,11,12,15)(10,14,13,16)(17,22,21,31,24,38,23,37)(18,28,27,40,30,47,29,46)(19,33,39,20,34,36,32,35)(25,42,48,26,43,45,41,44)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=hi,d^2=i,e^2=ghi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6737120);