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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1700 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,4)(5,7)(6,8)(9,10)(11,14)(12,13)(15,16)(17,18,22,21)(19,24,20,23)(25,26,29,27)(28,31,32,30);(1,3,5,8)(2,6,7,4)(10,12)(11,15)(17,19,22,20)(18,23,21,24);(3,8)(4,6)(9,10,13,12)(11,16,15,14)(17,18,22,21)(19,23,20,24)(25,27,29,26)(28,31,32,30);(1,4)(2,3)(5,6)(7,8)(9,11,13,15)(10,14,12,16)(17,20,22,19)(18,23,21,24)(25,28,29,32)(26,30,27,31);(1,2)(3,4)(5,7)(6,8)(9,12,13,10)(11,16,15,14)(17,21,22,18)(19,23,20,24)Polycyclic presentation:
2^9|a^2=fhi,b^2=fg,c^2=fghi,d^2=fghi,e^2=fgh,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, 1890032);