Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437547 |
| 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×#548 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1721 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)(3,6)(4,5)(7,8)(9,10)(11,14)(12,16)(13,15);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(17,18)(19,27)(20,25)(21,23)(22,26)(24,28)(29,31)(30,32);(9,12,13,11)(10,14,15,16)(18,23)(20,21)(24,27)(25,26)(28,30)(31,32);(17,19,22,29)(18,24,26,32)(20,28,21,30)(23,27,25,31);(1,3,4,7)(2,6,5,8)(17,20,22,21)(18,25,26,23)(19,28,29,30)(24,31,32,27)Polycyclic presentation:
2^9|b^2=g,c^2=g,d^2=i,e^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,h^b=hi,e^c=ei,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6354684);