Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #98134 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #379 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C22×C42 | C23 |
| prime radical | C24 | #120 |
| LCS | G ▷ C2×C42 ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C23 ▷ C1 | |
663 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,5)(3,7)(6,8)(10,14,15,16)(18,21,22,24);(1,2,4,6)(3,8,7,5)(9,10)(11,14)(12,16)(13,15)(19,23)(21,24);(9,11,13,12)(10,14,15,16)(17,18)(19,24)(20,22)(21,23)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=gh,d^2=h,e^2=i,b^a=bd,c^a=ce,d^a=dg,d^b=dh,e^b=ei,f^b=fgh,d^c=di,e^c=ei,f^c=fi71 conjugacy classes — view the full character table.
| maximal subgroup of | #72239137, #72239143, #72239149, … |
| maximal normal subgroup of | #72239137, #72239143, #72239149, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5341);