Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573485 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #36426 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#120 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2097 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(14,18)(19,27)(20,21)(22,28);(5,7,9,8)(6,10,12,11)(13,14)(15,22)(16,18)(17,20)(19,24)(21,25)(23,28)(26,27);(2,4)(5,7,9,8)(6,11,12,10)(13,15,16,23)(14,19,18,27)(17,26,25,24)(20,22,21,28);(5,7,9,8)(6,11,12,10)(13,14,16,18)(15,22,23,28)(17,20,25,21)(19,26,27,24);(5,8,9,7)(6,10,12,11)(14,20)(17,25)(18,21)(19,22)(24,26)(27,28)Polycyclic presentation:
2^9|b^2=fi,c^2=f,d^2=f,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=di,e^b=eh,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6486335);