Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667556 |
| 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×#459 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2402 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1411 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,7)(3,6,4,8)(9,10)(11,13)(12,15)(14,16)(19,26)(22,31)(27,32)(28,30);(1,3,5,4)(2,6,7,8)(9,11,13,10)(12,15,16,14)(17,18,20,24)(19,27,26,32)(21,23,29,25)(22,30,31,28);(1,4,5,3)(2,6,7,8)(10,11)(14,15)(17,19,20,26)(18,22,24,31)(21,30,29,28)(23,27,25,32);(9,12,13,16)(10,14,11,15)(19,26)(22,31)(27,32)(28,30);(18,23)(21,29)(22,27)(24,25)(28,30)(31,32)Polycyclic presentation:
2^9|a^2=fgi,b^2=g,c^2=fg,d^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6600681);