Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #96924 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C2×C22⋊C4 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1288 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,3)(9,10)(11,15,16,13)(12,14)(17,18,21,22,20,23,19,24);(1,2,4,3)(5,6)(7,8)(17,19,20,21)(18,22,23,24);(1,3,4,2)(6,8)(9,11)(10,13)(12,16)(14,15)(18,22)(19,21)(23,24)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=di,e^2=i,g^2=i,b^a=bd,c^a=ce,e^a=eh,c^b=cf,e^c=ei,g^c=ghi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 27306);