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 | #42454 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1383 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)(18,22)(19,25)(20,28)(23,29)(24,26)(27,31);(10,14)(13,16)(17,18,20,22,21,23,28,29)(19,26,25,32,27,24,31,30);(1,3,4,7)(2,6,5,8)(18,23)(22,29)(24,26)(30,32);(2,5)(6,8)(17,19,20,25,21,27,28,31)(18,24,22,30,23,26,29,32)Polycyclic presentation:
2^9|b^2=g,c^2=h,d^2=f,e^2=h,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 | #72313767, #72313813, #72313919 |
| maximal normal subgroup of | #72313767, #72313813, #72313919 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6719931);