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 | #42906 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1401 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(3,7)(9,10)(11,15)(12,14)(13,16)(17,18,23,29)(19,30,32,25)(20,26,35,43)(21,37,24,27)(22,38,39,28)(31,44,45,40)(33,46,47,41)(34,48,36,42);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19,23,32)(18,25,29,30)(20,33,35,47)(21,22,24,39)(26,41,43,46)(27,38,37,28)(31,34,45,36)(40,48,44,42);(10,14)(13,15)(17,20,21,34,23,35,24,36)(18,26,37,48,29,43,27,42)(19,31,39,47,32,45,22,33)(25,40,38,46,30,44,28,41);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,21,23,24)(18,27,29,37)(19,22,32,39)(20,34,35,36)(25,38,30,28)(26,42,43,48)(31,33,45,47)(40,46,44,41);(2,5)(6,8)(11,16)(13,15)(17,22,21,19,23,39,24,32)(18,28,37,25,29,38,27,30)(20,33,34,31,35,47,36,45)(26,41,48,40,43,46,42,44)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=hi,d^2=fg,e^2=hi,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^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6730670);