Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1258513 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2293 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,3)(9,10,14,11)(12,16,13,15)(17,18)(19,27)(20,29)(21,31)(22,23)(24,28)(25,30)(26,32);(6,8)(9,11,14,10)(12,16,13,15)(18,23)(19,28)(25,29)(26,31);(1,2,4,3)(5,6)(7,8)(9,12,14,13)(10,15,11,16)(17,19)(18,24)(20,26)(21,29)(22,28)(23,27)(25,32)(30,31);(9,12,14,13)(10,15,11,16)(17,20,22,30)(18,25,23,29)(19,26,28,31)(21,24,32,27);(1,3,4,2)(9,13,14,12)(10,16,11,15)(17,21,22,32)(18,26,23,31)(19,29,28,25)(20,27,30,24)Polycyclic presentation:
2^9|a^2=fi,b^2=fi,c^2=fg,d^2=i,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1218228);