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 | #392 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1457 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)(19,30)(20,36)(21,29)(22,41)(23,28)(24,34)(25,38)(26,40)(27,37)(31,42)(32,43)(33,44)(35,47)(39,45)(46,48);(9,11,12,16)(10,13,14,15)(17,19)(18,24)(20,37)(21,32)(22,38)(23,34)(25,27)(26,43)(28,30)(29,47)(31,39)(33,48)(35,40)(36,41)(42,44)(45,46);(2,5)(6,8)(10,14)(11,16)(17,20,21,39,23,38,40,33)(18,25,29,44,28,36,26,45)(19,31,35,37,34,48,32,22)(24,42,43,27,30,46,47,41);(10,14)(13,15)(17,21,23,40)(18,26,28,29)(19,32,34,35)(20,33,38,39)(22,31,37,48)(24,43,30,47)(25,44,36,45)(27,42,41,46);(1,3,4,7)(2,6,5,8)(17,22,40,48,23,37,21,31)(18,27,26,42,28,41,29,46)(19,33,32,38,34,39,35,20)(24,44,47,25,30,45,43,36)Polycyclic presentation:
2^9|b^2=fi,c^2=fhi,d^2=i,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, 6737765);