Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437418 |
| 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 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1670 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)(17,18,22,26)(19,27,28,24)(20,25,21,23)(29,31,30,32);(9,10,13,16)(11,15,12,14)(18,23)(19,28)(20,21)(24,32)(25,26)(27,31);(1,3,4,8)(2,5,6,7)(9,11,13,12)(10,14,16,15)(17,19)(18,24)(20,29)(21,30)(22,28)(23,31)(25,32)(26,27);(9,12,13,11)(10,15,16,14)(17,20,22,21)(18,25,26,23)(19,29,28,30)(24,32,27,31);(19,28)(24,27)(29,30)(31,32)Polycyclic presentation:
2^9|a^2=g,b^2=hi,c^2=gh,d^2=h,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,f^b=fi,e^c=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6375584);