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 | #392 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1473 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,14)(13,16)(17,18,24,30)(19,25,34,43)(20,26,36,45)(21,27,23,29)(22,28,35,44)(31,40,38,47)(32,41,39,48)(33,42,37,46);(9,11,12,16)(10,13,14,15)(17,19)(18,25)(20,35)(21,32)(22,36)(23,39)(24,34)(26,44)(27,41)(28,45)(29,48)(30,43)(31,33)(37,38)(40,42)(46,47);(2,5)(6,8)(10,14)(11,16)(17,20,23,33,24,36,21,37)(18,26,29,42,30,45,27,46)(19,31,32,35,34,38,39,22)(25,40,41,44,43,47,48,28);(10,14)(13,15)(17,21,24,23)(18,27,30,29)(19,32,34,39)(20,33,36,37)(22,38,35,31)(25,41,43,48)(26,42,45,46)(28,47,44,40);(1,3,4,7)(2,6,5,8)(17,22,23,31,24,35,21,38)(18,28,29,40,30,44,27,47)(19,33,32,36,34,37,39,20)(25,42,41,45,43,46,48,26)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=fh,d^2=i,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, 6737722);