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, socle | C24 | #120 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1162 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(5,6,7,8)(9,10,12,11,13,15,16,14)(17,18)(19,20,22,24)(21,23);(1,2)(3,4)(10,11,15,14)(17,19)(18,20,23,24)(21,22);(2,4)(9,11,12,10,13,14,16,15)(17,20)(18,22,23,19)(21,24)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=d,d^2=hi,f^2=hi,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=cf,d^b=di,f^b=fi,g^b=gi,e^c=eh,f^c=fh,g^c=gh80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 28032);