Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394746 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#124 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#394 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1200 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)(3,7)(4,8)(5,6)(9,10)(11,19)(12,15)(13,22)(14,18)(16,21)(17,23)(20,24);(2,6)(7,8)(9,11,13,12,14,20,23,21)(10,15,17,19,18,16,22,24);(1,3,5,4)(2,7,6,8)(9,12,13,20,14,21,23,11)(10,16,17,24,18,15,22,19);(1,4,5,3)(2,7,6,8)Polycyclic presentation:
2^9|b^2=eh,c^2=e,d^2=h,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,f^b=fi,f^c=fi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 323262);