Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2021 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| socle | C23 | C2×#120 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
2914 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(9,10)(11,15,19,14)(12,16)(13,17)(18,22,24,21)(20,23);(5,6)(7,8)(9,11)(10,14)(12,18)(13,19)(15,17)(16,21)(20,24)(22,23);(2,4)(6,8)(9,11)(10,15,17,14)(12,18)(13,19)(16,22,23,21)(20,24);(9,12,13,20)(10,16,17,23)(11,18,19,24)(14,21,15,22)Polycyclic presentation:
2^9|a^2=h,c^2=e,d^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,h^b=hi,f^c=fi,h^c=hi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72303776, #72303794 |
| maximal normal subgroup of | #72303776, #72303794 |
| maximal quotient of | #72303942 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265089);