Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13398 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #481 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C25 | SD16 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C23×M16 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
710 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(6,7)(10,14,15,22)(11,17)(12,20)(16,23,21,19)(18,24);(1,2)(3,4)(5,6)(7,8)(11,18)(14,22)(17,24)(19,23);(2,4)(5,7,8,6)(9,10,12,21,13,15,20,16)(11,19,17,14,18,23,24,22)Polycyclic presentation:
2^9|a^2=g,c^2=dh,e^2=h,h^2=i,b^a=bd,c^a=ce,e^a=eh,h^a=hi,c^b=cf,e^b=ei,d^c=di,e^c=eh,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55829);