Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519008 |
| 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×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1390 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,7,8,5)(9,10,13,12)(11,14,16,15)(17,18)(19,26)(20,25)(21,35)(22,37)(23,28)(24,31)(27,36)(29,40)(30,42)(32,39)(33,46)(34,47)(38,43)(41,45)(44,48);(1,3,4,8)(2,5,6,7)(9,11)(10,14)(12,15)(13,16)(17,19,22,31,23,30,36,21)(18,24,27,26,28,35,37,42)(20,29,34,43,32,44,45,33)(25,38,41,40,39,46,47,48);(2,6)(3,8)(9,12,13,10)(11,15,16,14)(17,20,23,32)(18,25,28,39)(19,29,30,44)(21,33,31,43)(22,34,36,45)(24,38,35,46)(26,40,42,48)(27,41,37,47);(2,6)(3,8)(11,16)(14,15)(19,30)(20,32)(21,31)(24,35)(25,39)(26,42)(34,45)(41,47);(2,6)(5,7)(17,21,36,30,23,31,22,19)(18,26,37,24,28,42,27,35)(20,33,45,44,32,43,34,29)(25,40,47,38,39,48,41,46)Polycyclic presentation:
2^9|a^2=hi,b^2=f,c^2=gh,e^2=fgi,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,d^c=di,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7514602);