Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806647 |
| 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 | #42471 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1402 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)(11,13)(12,14)(15,16);(1,2)(3,6)(4,5)(7,8)(9,11,12,15)(10,13,14,16)(17,18)(19,21)(20,25)(22,32)(23,27)(24,28)(26,30)(29,31);(10,14)(13,16)(17,19,22,28,23,29,20,26)(18,24,25,21,27,30,32,31);(1,3,4,7)(2,6,5,8)(17,20,23,22)(18,25,27,32)(19,28,29,26)(21,24,31,30);(2,5)(6,8)(17,21,20,24,23,31,22,30)(18,26,32,29,27,28,25,19)Polycyclic presentation:
2^9|b^2=g,c^2=h,d^2=fi,e^2=hi,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=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72313339, #72313698, #72313767 |
| maximal normal subgroup of | #72313339, #72313698, #72313767 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6719940);