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×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1265 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)(17,18,23,28)(19,30,32,24)(20,34,33,25)(21,29,35,26)(22,27,38,42)(31,44,45,39)(36,47,46,40)(37,43,48,41);(9,11)(10,13)(12,16)(14,15)(17,19,23,32)(18,24,28,30)(20,35,33,21)(22,36,38,46)(25,26,34,29)(27,40,42,47)(31,37,45,48)(39,43,44,41);(2,5)(6,8)(11,16)(13,15)(17,20,21,19,23,33,35,32)(18,25,29,24,28,34,26,30)(22,36,37,31,38,46,48,45)(27,40,43,39,42,47,41,44);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,21,23,35)(18,26,28,29)(19,20,32,33)(22,37,38,48)(24,34,30,25)(27,41,42,43)(31,36,45,46)(39,47,44,40);(17,22,21,37,23,38,35,48)(18,27,29,43,28,42,26,41)(19,31,33,46,32,45,20,36)(24,39,34,47,30,44,25,40)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fhi,d^2=g,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, 6733660);