Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907909 |
| 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×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2280 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1520 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,7,8,5)(17,18)(19,24)(20,31)(21,26)(22,36)(23,27)(25,33)(28,42)(29,38)(30,39)(32,41)(34,45)(35,40)(37,44)(43,48)(46,47);(1,3,4,8)(2,5,6,7)(9,10)(11,15)(12,14)(13,16)(17,19)(18,24)(20,28)(21,35)(22,30)(23,29)(25,37)(26,40)(27,38)(31,42)(32,46)(33,44)(34,43)(36,39)(41,47)(45,48);(3,8)(5,7)(9,11)(10,13)(12,16)(14,15)(17,20,22,33,23,34,21,32)(18,25,26,31,27,41,36,45)(19,28,30,44,29,43,35,46)(24,37,40,42,38,47,39,48);(2,6)(5,7)(11,16)(13,15)(19,29)(24,38)(28,43)(30,35)(37,47)(39,40)(42,48)(44,46);(10,14)(13,15)(17,21,23,22)(18,26,27,36)(19,30,29,35)(20,32,34,33)(24,39,38,40)(25,31,41,45)(28,44,43,46)(37,48,47,42)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=g,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efhi,d^c=dfh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6816486);