Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476297 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
758 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)(9,10,12,18,14,16,13,17)(11,19,21,24,22,15,20,23);(1,3,5,4)(2,6,7,8)(9,11)(10,15)(12,20)(13,21)(14,22)(16,19)(17,23)(18,24);(1,4,5,3)(2,6,7,8)(10,16)(11,20)(12,13)(15,23)(19,24)(21,22);(3,4)(6,8)(10,17,16,18)(15,23,19,24)Polycyclic presentation:
2^9|a^2=g,b^2=ef,c^2=ef,d^2=f,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=def,g^b=gi,d^c=de,g^c=gi,h^c=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 386642);