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 | #2388 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1446 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)(3,5)(4,6)(7,8)(9,10)(11,15)(12,14)(13,16)(17,18,23,28)(19,24,31,38)(20,25,29,36)(21,26,32,39)(22,27,35,42)(30,37,44,47)(33,40,43,46)(34,41,45,48);(1,3,4,7)(2,5,6,8)(9,11)(10,13)(12,16)(14,15)(17,19)(18,24)(20,32)(21,29)(22,33)(23,31)(25,39)(26,36)(27,40)(28,38)(30,34)(35,43)(37,41)(42,46)(44,45)(47,48);(2,6)(5,8)(11,16)(13,15)(17,20,32,19,23,29,21,31)(18,25,39,24,28,36,26,38)(22,33,45,30,35,43,34,44)(27,40,48,37,42,46,41,47);(10,14)(13,15)(17,21,23,32)(18,26,28,39)(19,29,31,20)(22,34,35,45)(24,36,38,25)(27,41,42,48)(30,43,44,33)(37,46,47,40);(17,22,32,45,23,35,21,34)(18,27,39,48,28,42,26,41)(19,30,29,43,31,44,20,33)(24,37,36,46,38,47,25,40)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=fhi,d^2=i,e^2=fhi,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, 6720697);