Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1258513 |
| 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 | |
1728 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)(3,7)(4,6)(5,8)(10,12)(14,15)(17,18,21,24)(19,23,20,22);(1,3)(2,4)(5,6)(7,8)(25,26,29,28)(27,31,32,30);(1,4)(2,6)(3,8)(5,7)(9,10,13,12)(11,14,16,15)(17,19,21,20)(18,22,24,23)(25,27,29,32)(26,30,28,31);(1,4)(2,3)(5,7)(6,8)(9,11,13,16)(10,14,12,15)(17,18)(19,22)(20,23)(21,24);(1,3)(2,7)(4,8)(5,6)(9,12,13,10)(11,15,16,14)(17,20,21,19)(18,23,24,22)(25,28,29,26)(27,31,32,30)Polycyclic presentation:
2^9|a^2=fi,b^2=fh,c^2=fgh,d^2=fgi,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1209868);