Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437412 |
| 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 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1734 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)(17,18,21,23)(19,25,26,32)(20,28,30,24)(22,29,31,27);(9,10,13,16)(11,15,12,14)(17,19,21,26)(18,22,23,31)(20,29,30,27)(24,32,28,25);(1,3,4,8)(2,5,6,7)(9,11,13,12)(10,14,16,15)(18,23)(19,26)(24,28)(27,29);(9,12,13,11)(10,15,16,14)(19,26)(22,31)(25,32)(27,29);(18,23)(22,31)(24,28)(25,32)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=gh,d^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,c^b=ch,d^b=dh,f^b=fi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6375580);