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 | #15412 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2772 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)(9,10,12,13)(11,15,16,14)(17,18)(19,22)(20,24)(21,29)(23,26)(25,31)(27,32)(28,30);(6,8)(10,13)(11,16)(17,19,23,20)(18,24,26,22)(21,27,31,28)(25,32,29,30);(5,6)(7,8)(10,13)(14,15)(17,20,23,19)(18,24,26,22)(21,27,31,28)(25,30,29,32);(1,2,3,4)(9,10)(11,14)(12,13)(15,16)(17,21)(18,25)(19,27)(20,28)(22,30)(23,31)(24,32)(26,29);(9,11,12,16)(10,14,13,15)(17,22,23,24)(18,19,26,20)(21,30,31,32)(25,27,29,28)Polycyclic presentation:
2^9|a^2=g,b^2=fg,c^2=fg,d^2=fi,e^2=f,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, 1487992);