Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7513980 |
| 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 | #26616 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#126 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1500 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)(3,7)(4,8)(5,6)(9,10)(11,15)(12,13)(14,16)(17,18)(19,25)(20,23)(21,24)(22,26)(27,30)(28,32)(29,31);(2,6)(7,8)(10,13)(11,16)(17,19)(18,22)(20,26)(21,27)(23,25)(24,32)(28,29)(30,31);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,14,13,15)(18,23)(19,26)(24,31)(27,28);(10,13)(14,15)(18,23)(19,27)(21,29)(22,30)(25,32)(26,28);(1,4,5,3)(2,7,6,8)(10,13)(14,15)(18,23)(22,25)(24,31)(30,32)Polycyclic presentation:
2^9|c^2=fi,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,h^d=hi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7483860);