Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573477 |
| 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 | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2090 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6,9,10)(7,11,8,12)(13,14)(15,20)(16,21)(17,19)(18,22)(23,27)(24,25)(26,28);(6,10)(11,12)(14,19)(15,21)(16,25)(18,27)(20,26)(24,28);(2,4)(13,15)(14,20)(16,23)(17,24)(18,28)(19,25)(21,27)(22,26);(5,7,9,8)(6,11,10,12)(14,19)(15,21)(16,25)(18,27)(20,26)(24,28);(5,8,9,7)(6,12,10,11)(13,16,17,26)(14,21,19,28)(15,23,24,22)(18,20,27,25)Polycyclic presentation:
2^9|a^2=fi,d^2=fi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=eh,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6485349);