Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | #392 | C2×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1379 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,5)(3,7)(9,10)(11,13)(12,14)(15,16)(17,18,23,26,22,21,20,29)(19,25,27,31,28,24,30,32);(1,2,4,5)(3,6,7,8)(9,11,12,15)(10,13,14,16)(17,19,22,28)(18,24,21,25)(20,30,23,27)(26,32,29,31);(2,5)(6,8)(10,14)(13,16)(17,19)(18,25)(20,30)(21,24)(22,28)(23,27)(26,31)(29,32);(1,3,4,7)(2,6,5,8)(17,20,22,23)(18,26,21,29)(19,27,28,30)(24,31,25,32);(17,21,23,29,22,18,20,26)(19,25,27,31,28,24,30,32)Polycyclic presentation:
2^9|a^2=h,b^2=fg,d^2=f,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=cf,d^b=di,d^c=di,e^c=ei,e^d=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6285954);