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 | |
1483 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)(20,25)(21,29)(22,27)(23,28)(24,34)(26,39)(31,46)(32,41)(33,44)(35,47)(36,45)(37,43)(38,42)(40,48);(9,11,12,16)(10,13,14,15)(17,19)(18,24)(20,36)(21,32)(22,37)(23,34)(25,27)(26,41)(28,30)(29,47)(31,38)(33,48)(35,39)(40,42)(43,45)(44,46);(2,5)(6,8)(10,14)(11,16)(17,20,21,38,23,37,39,33)(18,25,29,42,28,43,26,44)(19,31,35,36,34,48,32,22)(24,40,41,27,30,46,47,45);(10,14)(13,15)(17,21,23,39)(18,26,28,29)(19,32,34,35)(20,33,37,38)(22,31,36,48)(24,41,30,47)(25,42,43,44)(27,40,45,46);(1,3,4,7)(2,6,5,8)(17,22,39,48,23,36,21,31)(18,27,26,40,28,45,29,46)(19,33,32,37,34,38,35,20)(24,42,47,25,30,44,41,43)Polycyclic presentation:
2^9|b^2=fi,c^2=fhi,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=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6737735);