Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2031997 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C24×C42 |
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 | |
2643 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)(3,7)(4,5)(6,8)(9,10)(11,15)(12,16)(13,14)(17,18,20,22)(19,23,24,21)(25,26)(27,31)(28,32)(29,30);(2,4)(6,7)(10,13)(15,16)(19,24)(21,23)(26,29)(27,32);(9,11,14,12)(10,15,13,16)(17,18)(19,21)(20,22)(23,24)(25,27,30,32)(26,31,29,28);(1,3,5,8)(2,6,4,7)(9,12,14,11)(10,16,13,15)(17,19,20,24)(18,21,22,23)(25,28)(26,27)(29,32)(30,31);(1,4,5,2)(3,7,8,6)(9,13,14,10)(11,16,12,15)(17,18)(19,21)(20,22)(23,24)(25,29,30,26)(27,28,32,31)Polycyclic presentation:
2^9|a^2=g,c^2=fghi,d^2=fgh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1959557);