Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806575 |
| 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×#127 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#127 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1424 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,5)(3,7)(9,10)(11,14)(12,16)(13,15)(19,25)(22,30)(26,32)(28,31);(1,2)(3,6)(4,5)(7,8)(9,11,13,12)(10,14,15,16)(17,18,20,23)(19,26,25,32)(21,29,27,24)(22,31,30,28);(9,12,13,11)(10,14,15,16)(17,19,27,31,20,25,21,28)(18,22,29,32,23,30,24,26);(1,3,4,7)(2,6,5,8);(2,5)(6,8)(17,19,27,31,20,25,21,28)(18,22,29,32,23,30,24,26)Polycyclic presentation:
2^9|b^2=g,c^2=gh,d^2=f,e^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6726256);