Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806839 |
| 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 | #30549 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2401 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1567 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,7)(3,8,4,6)(9,10)(11,14)(12,13)(15,16)(17,18,23,29)(19,30,34,25)(20,35,38,26)(21,39,24,27)(22,41,36,28)(31,43,40,48)(32,44,42,46)(33,45,37,47);(1,2)(3,6)(4,8)(5,7)(17,19,23,34)(18,25,29,30)(20,36,38,22)(21,32,24,42)(26,28,35,41)(27,44,39,46)(31,37,40,33)(43,45,48,47);(10,13)(14,16)(17,20,21,37,23,38,24,33)(18,26,39,45,29,35,27,47)(19,31,42,22,34,40,32,36)(25,43,44,41,30,48,46,28);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,14,13,16)(17,21,23,24)(18,27,29,39)(19,32,34,42)(20,37,38,33)(22,31,36,40)(25,44,30,46)(26,47,35,45)(28,43,41,48);(1,4,5,3)(2,6,7,8)(17,22,24,31,23,36,21,40)(18,28,27,48,29,41,39,43)(19,33,32,38,34,37,42,20)(25,45,46,26,30,47,44,35)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=hi,d^2=fgi,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6735232);