Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
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 | |
2179 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,6)(3,7)(10,13)(11,15)(17,18,20,22)(19,23,24,21);(1,2,5,6)(3,4,7,8)(9,10,12,13)(11,16,15,14)(17,18,20,22)(19,21,24,23)(25,26,28,30)(27,29,31,32);(2,6)(3,7)(9,10,12,13)(11,14,15,16)(17,18)(19,21)(20,22)(23,24)(25,26)(27,29)(28,30)(31,32);(1,3)(2,4)(5,7)(6,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,21)(20,22)(23,24);(1,4)(2,3)(5,8)(6,7)(9,11,12,15)(10,14,13,16)(17,19)(18,21)(20,24)(22,23)(25,27,28,31)(26,29,30,32)Polycyclic presentation:
2^9|a^2=fi,b^2=fghi,c^2=fgi,e^2=hi,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, 1806853);