Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #115980 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1439 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C22×C42 | C23 |
| prime radical | C24 | C4×D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
711 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(3,7)(5,8)(9,10,12,16,13,15,21,23)(11,17,18,24,19,14,20,22);(2,5,6,8);(1,2)(3,8)(4,6)(5,7)(9,11,13,19)(10,14,15,17)(12,20,21,18)(16,24,23,22)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=i,f^2=h,g^2=i,b^a=bd,c^a=ce,e^a=ei,f^a=fh,c^b=cf,d^c=dh,f^c=fh,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 23180);