Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806647 |
| 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 | #31030 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1506 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)(11,14)(12,13)(15,16)(17,18,24,30)(19,25,33,41)(20,26,31,39)(21,27,23,29)(22,28,38,46)(32,40,47,48)(34,42,36,44)(35,43,37,45);(2,5)(3,7)(17,19,24,33)(18,25,30,41)(20,21,31,23)(22,36,38,34)(26,27,39,29)(28,44,46,42)(32,35,47,37)(40,43,48,45);(10,13)(14,16)(17,20,23,19,24,31,21,33)(18,26,29,25,30,39,27,41)(22,36,37,47,38,34,35,32)(28,44,45,48,46,42,43,40);(2,5)(6,8)(9,11,12,15)(10,14,13,16)(17,21,24,23)(18,27,30,29)(19,31,33,20)(22,35,38,37)(25,39,41,26)(28,43,46,45)(32,36,47,34)(40,44,48,42);(1,3,4,7)(2,6,5,8)(17,22,21,35,24,38,23,37)(18,28,27,43,30,46,29,45)(19,32,20,34,33,47,31,36)(25,40,26,42,41,48,39,44)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=h,d^2=gi,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=di,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6732608);