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×#550 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1511 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)(3,7)(4,8)(5,6)(9,10)(11,15)(12,13)(14,16)(17,18)(19,21)(20,28)(22,25)(23,26)(24,30)(27,29)(31,32);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,23,25)(18,21,26,22)(20,24,29,32)(27,31,28,30);(10,13)(14,15)(17,20,22,31,23,29,21,30)(18,24,19,27,26,32,25,28);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,14,13,15)(17,19,22,26,23,25,21,18)(20,27,31,32,29,28,30,24);(1,4,5,3)(2,7,6,8)(20,29)(24,32)(27,28)(30,31)Polycyclic presentation:
2^9|b^2=i,c^2=h,d^2=fgi,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7505227);