Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667836 |
| 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 | #1999 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1510 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)(11,15)(12,16)(13,14)(17,18,22,25)(19,23,30,26)(20,24,31,21)(27,29,32,28);(1,3,4,8)(2,5,6,7)(17,19,22,30)(18,23,25,26)(20,27,31,32)(21,28,24,29);(10,14)(15,16)(17,20,22,31)(18,24,25,21)(19,26,30,23)(27,29,32,28);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(17,20)(18,24)(19,27)(21,25)(22,31)(23,29)(26,28)(30,32);(3,8)(5,7)(9,12,13,11)(10,16,14,15)(19,28)(20,31)(21,24)(23,27)(26,32)(29,30)Polycyclic presentation:
2^9|a^2=fi,b^2=fi,c^2=i,d^2=g,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6638044);