Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438997 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1366 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,14)(12,13)(15,16)(18,23)(20,29)(24,31)(25,26);(1,3,4,8)(2,5,6,7)(17,18,21,23)(19,25,27,26)(20,22,29,30)(24,32,31,28);(10,13)(14,16)(17,19,22,28,21,27,30,32)(18,24,20,25,23,31,29,26);(17,20,22,23,21,29,30,18)(19,26,28,24,27,25,32,31);(9,11,12,15)(10,14,13,16)(19,27)(24,31)(25,26)(28,32)Polycyclic presentation:
2^9|a^2=fi,b^2=f,c^2=h,d^2=h,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=dh,h^b=hi,d^c=di,e^c=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6375181);