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 | #45437 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1463 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)(9,10,13,12)(11,14,16,15)(17,18)(19,28)(20,23)(21,24)(22,26)(25,32)(27,31)(29,30);(2,6)(7,8)(9,11)(10,14)(12,15)(13,16)(17,19)(18,24)(20,30)(21,22)(23,32)(25,27)(26,28)(29,31);(1,3,5,4)(2,7,6,8)(9,12,13,10)(11,15,16,14)(17,20,31,26,22,27,23,18)(19,24,32,25,21,28,29,30);(1,3,5,4)(2,7,6,8)(11,16)(14,15)(17,21,31,29,22,19,23,32)(18,25,20,28,26,30,27,24);(1,4,5,3)(2,7,6,8)(18,26)(20,27)(24,28)(25,30)Polycyclic presentation:
2^9|a^2=ghi,c^2=fh,d^2=fg,e^2=fi,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, 7512868);