Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9914 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C23 | #386 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
899 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 28 points:
(5,6)(7,10,12,9)(8,11)(14,19,22,21)(20,27,28,23);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(13,14)(15,23)(16,21)(17,19)(18,22)(20,25)(24,28)(26,27);(2,4)(5,7)(6,10,11,9)(8,12)(13,15,16,24,18,26,17,25)(14,20,19,27,22,28,21,23)Polycyclic presentation:
2^9|a^2=g,c^2=d,d^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fh,c^b=cf,d^b=di,e^b=eh,f^b=fi,g^b=ghi,e^c=eh,g^c=gh80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 26461);