Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518964 |
| 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 | C22×D4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1317 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)(19,26)(20,29)(21,31)(22,32)(23,25)(24,28)(27,30);(2,6)(7,8)(9,11)(10,14)(12,15)(13,16)(17,19,23,28)(18,24,25,26)(20,22,27,31)(21,30,32,29);(1,3,5,4)(2,7,6,8)(9,12,13,10)(11,15,16,14)(17,20,22,28,23,27,31,19)(18,24,21,29,25,26,32,30);(1,3,5,4)(2,7,6,8)(11,16)(14,15)(17,21,31,18,23,32,22,25)(19,26,27,29,28,24,20,30);(1,4,5,3)(2,7,6,8)(17,22,23,31)(18,21,25,32)(19,27,28,20)(24,30,26,29)Polycyclic presentation:
2^9|a^2=ghi,b^2=i,c^2=fhi,d^2=fg,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,d^c=di,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7512832);