Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8674 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #380 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#380 |
| prime radical | #3444 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
978 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,8,6,3,4,7)(9,10,14,17,15,12,13,23)(11,18,21,19,22,24,20,16);(9,11,13,20,15,22,14,21)(10,16,17,18,12,19,23,24);(1,3)(2,6)(4,8)(5,7)(9,12)(10,15)(11,19)(13,23)(14,17)(16,22)(18,21)(20,24);(1,4,6,5)(2,7,3,8)Polycyclic presentation:
2^9|a^2=g,b^2=e,d^2=i,e^2=h,f^2=hi,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,f^c=fhi,g^c=gh92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 96619);