Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| 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 | C22×#121 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2616 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,4)(9,10,13,16)(11,15,12,14)(17,18,21,24)(19,25,27,22)(20,28,30,23)(26,31,29,32);(6,8)(11,12)(14,15)(17,19,21,27)(18,22,24,25)(20,29,30,26)(23,32,28,31);(5,6)(7,8)(9,11,13,12)(10,14,16,15)(17,19)(18,22)(20,26)(21,27)(23,31)(24,25)(28,32)(29,30);(1,2,3,4)(9,12,13,11)(10,15,16,14)(17,18)(19,25)(20,23)(21,24)(22,27)(26,32)(28,30)(29,31);(9,10)(11,14)(12,15)(13,16)(17,20)(18,23)(19,26)(21,30)(22,31)(24,28)(25,32)(27,29)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=fg,d^2=fi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1487968);