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 | #3061 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1569 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)(17,18,24,27)(19,25,31,28)(20,22,34,23)(21,26,38,43)(29,40,45,44)(30,41,39,32)(33,36,47,37)(35,42,48,46);(1,3,6,12)(2,7,9,10)(4,11,5,14)(8,15,13,16)(17,19)(18,25)(20,32)(21,35)(22,30)(23,39)(24,31)(26,42)(27,28)(29,37)(33,40)(34,41)(36,45)(38,48)(43,46)(44,47);(17,20,23,27,24,34,22,18)(19,28,30,32,31,25,39,41)(21,33,37,43,38,47,36,26)(29,44,35,46,45,40,48,42);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,10,16);(1,5,6,4)(2,8,9,13)(3,11,12,14)(7,16,10,15)(17,21,22,36,24,38,23,37)(18,26,34,47,27,43,20,33)(19,29,39,48,31,45,30,35)(25,40,32,46,28,44,41,42)Polycyclic presentation:
2^9|a^2=fi,b^2=f,c^2=h,d^2=f,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, 6719063);