Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13331 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #459 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C22×#509 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
1102 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,4)(5,6,7,8)(10,14)(11,18)(12,19)(15,21)(16,22)(17,23)(20,24);(2,4)(9,10)(11,15)(12,21)(13,16)(14,18)(17,19)(20,23)(22,24);(10,15)(11,18)(12,19)(14,17)(16,23)(20,24)(21,22)Polycyclic presentation:
2^9|a^2=g,d^2=hi,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fh,h^a=hi,c^b=cf,d^b=dh,e^b=ei,f^b=fh,h^b=hi,d^c=dh,f^c=fh,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72250557, #72250663, #72250666, … |
| maximal normal subgroup of | #72250557, #72250663, #72250666, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55131);