Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #15267 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2536 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,5)(3,7,8,6)(10,13)(14,15)(17,18,21,19)(20,23,24,22)(25,26)(27,28)(29,31)(30,32);(2,5)(3,8)(9,10)(11,15)(12,13)(14,16)(17,18)(19,21)(20,23)(22,24)(25,27,29,32)(26,30,31,28);(2,5)(6,7)(10,13)(14,15)(18,19)(22,23)(25,28,29,30)(26,27,31,32);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,21,18)(20,23,24,22)(25,28)(26,27)(29,30)(31,32);(1,3,4,8)(2,6,5,7)(9,11,12,16)(10,14,13,15)(17,20,21,24)(18,22,19,23)(25,26)(27,28)(29,31)(30,32)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=ghi,d^2=fgh,e^2=fghi,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, 1512565);