Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500669 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1812 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,6,7)(3,5,8,4)(9,10)(11,13)(12,14)(15,16)(17,18,22,26)(19,28,30,24)(20,25,35,41)(21,36,23,27)(29,45,46,38)(31,39,32,44)(33,48,47,40)(34,43,37,42);(1,3,6,8)(2,4,7,5)(9,11,12,15)(10,13,14,16)(17,19,22,30)(18,24,26,28)(20,33,35,47)(21,32,23,31)(25,40,41,48)(27,44,36,39)(29,37,46,34)(38,42,45,43);(1,4,6,5)(2,3,7,8)(10,14)(13,16)(17,20,22,35)(18,25,26,41)(19,29,30,46)(21,34,23,37)(24,38,28,45)(27,42,36,43)(31,33,32,47)(39,48,44,40);(1,5,6,4)(2,3,7,8)(10,14)(13,16)(18,26)(19,30)(25,41)(27,36)(29,46)(31,32)(33,47)(42,43);(17,21,22,23)(18,27,26,36)(19,31,30,32)(20,34,35,37)(24,39,28,44)(25,42,41,43)(29,33,46,47)(38,48,45,40)Polycyclic presentation:
2^9|a^2=f,b^2=fg,c^2=f,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,e^b=ei,h^b=hi,h^c=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7412556);