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 | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1481 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)(9,10,12,14)(11,15,16,13)(17,18,24,30)(19,25,32,39)(20,26,34,41)(21,27,23,29)(22,28,37,44)(31,38,45,47)(33,40,46,48)(35,42,36,43);(2,5)(3,7)(17,19)(18,25)(20,21)(22,33)(23,34)(24,32)(26,27)(28,40)(29,41)(30,39)(31,36)(35,45)(37,46)(38,43)(42,47)(44,48);(9,11,12,16)(10,13,14,15)(17,20,21,19,24,34,23,32)(18,26,27,25,30,41,29,39)(22,33,35,31,37,46,36,45)(28,40,42,38,44,48,43,47);(2,5)(6,8)(17,21,24,23)(18,27,30,29)(19,20,32,34)(22,35,37,36)(25,26,39,41)(28,42,44,43)(31,33,45,46)(38,40,47,48);(1,3,4,7)(2,6,5,8)(17,22,23,36,24,37,21,35)(18,28,29,43,30,44,27,42)(19,31,20,33,32,45,34,46)(25,38,26,40,39,47,41,48)Polycyclic presentation:
2^9|a^2=gi,c^2=ghi,d^2=i,e^2=fh,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, 6725358);