Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #238422 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#393 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1075 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)(4,6)(5,8)(9,10)(11,14)(12,19,21,15)(13,16)(17,22,20,24)(18,23);(3,4)(6,7)(9,11)(10,14)(12,20,21,17)(13,18)(15,24,19,22)(16,23);(2,6)(3,4)(7,8)(9,12)(10,15)(11,17)(13,21)(14,22)(16,19)(18,20)(23,24);(1,3,5,4)(2,7,8,6)(12,21)(15,19)(17,20)(22,24)Polycyclic presentation:
2^9|a^2=e,b^2=e,d^2=hi,f^2=hi,b^a=be,c^a=cf,f^a=fi,g^a=ghi,c^b=cg,d^b=dhi,f^b=fi,g^b=gh,d^c=di,e^c=eh,f^c=fhi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 145622);