Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2805613 |
| automorphism | Aut(G) order 1048576 |
| 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 | |
2330 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)(9,10,13,14)(11,15,16,12)(17,18)(19,22)(20,28)(21,25)(23,27)(24,29)(26,31)(30,32);(6,8)(10,14)(12,15)(17,19,21,27)(18,22,25,23)(20,26,29,32)(24,30,28,31);(5,6)(7,8)(17,19)(18,23)(20,26)(21,27)(22,25)(24,31)(28,30)(29,32);(1,2)(3,4)(9,11,13,16)(10,12,14,15)(17,18)(19,23)(20,24)(21,25)(22,27)(26,31)(28,29)(30,32);(9,12)(10,11)(13,15)(14,16)(17,20,21,29)(18,24,25,28)(19,26,27,32)(22,30,23,31)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2713385);