Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #504125 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#476 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #3050 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C8×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
567 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6,10,13,11,9,8,14)(7,12,19,20,17,18,15,16);(2,4)(5,7,8,15,11,17,10,19)(6,12,14,16,9,18,13,20);(6,13)(7,12,17,18)(8,10)(9,14)(15,20,19,16);(2,4)Polycyclic presentation:
2^9|a^2=fi,b^2=f,c^2=fh,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,h^b=hi,f^c=fi,g^c=gh,h^c=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72252778, #72252782, #72252800, … |
| maximal normal subgroup of | #72252778, #72252782, #72252800, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 411323);