Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6884841 |
| 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 | #30512 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1391 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,6)(4,8)(5,7)(17,18,24,28,23,30,22,29)(19,25,20,27,31,26,21,32);(1,2,6,3)(4,7,5,8)(9,10)(11,15)(12,14)(13,16)(17,19,23,31)(18,25,30,26)(20,22,21,24)(27,29,32,28);(1,3,6,2)(4,7,5,8)(9,11)(10,13)(12,16)(14,15)(17,20)(18,26)(19,24)(21,23)(22,31)(25,30)(27,28)(29,32);(1,4,6,5)(2,7,3,8)(11,16)(13,15)(17,21,23,20)(18,27,30,32)(19,22,31,24)(25,28,26,29);(1,5,6,4)(2,7,3,8)(10,14)(13,15)(17,22,23,24)(18,28,30,29)(19,20,31,21)(25,32,26,27)Polycyclic presentation:
2^9|a^2=fgi,b^2=fi,c^2=f,d^2=fi,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dfi,e^a=ei,c^b=ch,d^b=di,e^b=efh,d^c=dfh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6815040);