Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806815 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #60695 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1587 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,6,9)(3,10,12,7)(4,13,5,8)(11,15,14,16)(17,18,22,25)(19,29,31,24)(20,33,32,43)(21,26,36,39)(23,28,34,27)(30,44,45,38)(35,47,46,40)(37,42,48,41);(1,3,6,12)(2,7,9,10)(4,11,5,14)(8,15,13,16)(17,19,22,31)(18,24,25,29)(20,34,32,23)(21,35,36,46)(26,40,39,47)(27,33,28,43)(30,37,45,48)(38,42,44,41);(17,20,23,19,22,32,34,31)(18,24,28,33,25,29,27,43)(21,35,37,30,36,46,48,45)(26,38,42,47,39,44,41,40);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,10,16)(18,25)(24,29)(26,39)(27,28)(33,43)(38,44)(40,47)(41,42);(1,5,6,4)(2,8,9,13)(3,11,12,14)(7,16,10,15)(17,21,23,37,22,36,34,48)(18,26,28,42,25,39,27,41)(19,30,32,46,31,45,20,35)(24,38,33,47,29,44,43,40)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=h,d^2=fi,e^2=fhi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6719066);