Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417728 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2424 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1055 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,7)(3,6,4,8)(9,10,13,17,14,11,18,19)(12,15,21,20,23,16,22,24);(1,3,5,4)(2,6,7,8)(9,11)(10,14)(12,20)(13,17)(15,21)(16,22)(18,19)(23,24);(2,7)(6,8)(9,12)(10,15)(11,16)(13,21)(14,23)(17,20)(18,22)(19,24);(1,4,5,3)(2,8,7,6)(9,12)(10,16)(11,15)(13,22)(14,23)(17,20)(18,21)(19,24)Polycyclic presentation:
2^9|a^2=ef,b^2=f,d^2=f,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,g^c=gi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 337129);