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 | C2×#449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1338 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,5)(3,6,7,8)(9,10,12,14)(11,15,16,13)(17,18)(19,24)(20,32)(21,26)(22,27)(23,28)(25,29)(30,37)(31,38)(33,39)(34,42)(35,41)(36,43)(40,44)(45,47)(46,48);(9,11)(10,13)(12,16)(14,15)(17,19)(18,24)(20,33)(21,29)(22,34)(23,31)(25,26)(27,42)(28,38)(30,35)(32,39)(36,44)(37,41)(40,43)(45,46)(47,48);(2,5)(6,8)(10,14)(11,16)(17,20,33,19,23,29,21,31)(18,25,39,38,28,32,26,24)(22,34,46,30,36,44,35,45)(27,40,48,47,43,42,41,37);(10,14)(13,15)(17,21,23,33)(18,26,28,39)(19,29,31,20)(22,35,36,46)(24,25,38,32)(27,41,43,48)(30,44,45,34)(37,40,47,42);(1,3,4,7)(2,6,5,8)(17,22,33,46,23,36,21,35)(18,27,39,48,28,43,26,41)(19,30,29,44,31,45,20,34)(24,37,25,40,38,47,32,42)Polycyclic presentation:
2^9|a^2=gi,c^2=fhi,d^2=i,e^2=gh,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, 6739694);