Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24264 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C23×D4 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1706 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,14)(12,13)(15,16);(1,3,4,8)(2,5,6,7)(18,21)(19,22)(23,24);(10,13)(14,16)(17,18,22,24,20,23,19,21);(17,18,22,24,20,23,19,21);(9,11,12,15)(10,14,13,16)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=hi,d^2=hi,e^2=g,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,h^b=hi116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6375124);