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 | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1922 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,12)(13,14)(15,16)(17,18,21,23)(19,20,24,22)(25,26,29,28)(27,31,32,30);(1,3)(2,6)(4,7)(5,8)(10,14)(12,15)(17,18)(19,22)(20,24)(21,23)(25,26,29,28)(27,30,32,31);(3,8)(6,7)(9,11,13,16)(10,12,14,15)(17,19,21,24)(18,22,23,20)(25,26,29,28)(27,30,32,31);(1,3)(2,7)(4,6)(5,8)(9,11)(10,15)(12,14)(13,16)(17,19)(18,20)(21,24)(22,23)(25,27,29,32)(26,30,28,31);(1,4,5,2)(3,6,8,7)(9,12,13,15)(10,11,14,16)(17,20)(18,19)(21,22)(23,24)(25,28,29,26)(27,31,32,30)Polycyclic presentation:
2^9|a^2=fi,b^2=fghi,c^2=gi,d^2=fghi,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, 1806837);