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 | #28484 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1591 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,7)(3,4,8,5)(9,10,12,13)(11,15,16,14)(17,18,22,27)(19,28,29,24)(20,30,31,25)(21,26,23,32);(1,3)(2,5)(4,7)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,29)(18,24,27,28)(20,21,31,23)(25,26,30,32);(1,3)(2,4)(5,7)(6,8)(10,13)(14,15)(17,20,22,31)(18,24,27,28)(19,21,29,23)(25,32,30,26);(1,4,6,5)(2,3,7,8)(9,11,12,16)(10,14,13,15)(17,20)(18,25)(19,21)(22,31)(23,29)(24,26)(27,30)(28,32);(1,5,6,4)(2,3,7,8)(17,21,22,23)(18,26,27,32)(19,20,29,31)(24,25,28,30)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,d^2=fi,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,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, 6872035);