Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #188761 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#117 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#381 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3790 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6)(7,14,16,10)(8,13)(9,12)(11,15,18,20)(17,19);(5,7)(6,10)(8,15)(9,16)(11,13)(12,14)(17,20)(18,19);(5,7)(6,11,12,18)(8,15)(9,16)(10,13,14,19)(17,20);(2,4)Polycyclic presentation:
2^9|a^2=h,c^2=e,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,h^b=hi,h^c=hi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72257017, #72257147, #72257172, … |
| maximal normal subgroup of | #72257017, #72257147, #72257172, … |
| maximal quotient of | #72303054, #72303063, #72303078, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299425);