Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #483919 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#442 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | #3050 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
582 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)(3,6)(4,5)(7,8)(9,10,12,18,14,19,13,23)(11,15,20,24,22,16,21,17);(2,5)(3,7)(9,11,12,20,14,22,13,21)(10,15,18,24,19,16,23,17);(1,3,4,7)(2,6,5,8)(10,16)(11,20)(12,13)(15,19)(17,23)(18,24)(21,22);(2,5)(6,8)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=e,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=ceh,f^b=fi,f^c=fhi,g^c=gi,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72265122, #72265177, #72265181, … |
| maximal normal subgroup of | #72265122, #72265177, #72265181, … |
| maximal quotient of | #72309868, #72309878, #72309994, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 404080);