Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393906 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2404 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1223 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)(3,8)(4,6)(5,7)(15,22)(21,24);(1,3,5,4)(2,6,7,8)(10,14,16,19)(15,21,22,24);(9,10)(11,15)(12,19)(13,16)(14,20)(17,22)(18,24)(21,23);(1,4,5,3)(2,6,7,8)(9,11)(10,15)(12,18)(13,17)(14,21)(16,22)(19,24)(20,23)Polycyclic presentation:
2^9|b^2=eh,d^2=e,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,g^c=gi,h^c=hi,f^d=fi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 312881);