Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #125508 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#102 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | C2×SD16 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
416 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 416 conjugacy classes of subgroups — too many to draw. Only its 98 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 52 points:
(1,2)(3,4)(5,6,9,13,10,14,8,12)(7,15,16,19,18,20,17,11)(21,22,25,30,26,31,40,48,27,32,41,49,39,47,24,29)(23,33,35,45,36,46,34,50,37,51,38,28,42,43,52,44);(5,7)(6,11,14,19)(8,17)(9,16)(10,18)(12,20,13,15)(21,23,26,42,27,37,39,36)(22,28,31,50,32,45,47,44)(24,38,25,34,40,35,41,52)(29,46,30,33,48,43,49,51);(2,4)(5,8,10,9)(6,12,14,13)(7,16,18,17)(11,15,19,20)Polycyclic presentation:
2^9|a^2=f,b^2=dfh,c^2=g,d^2=ghi,f^2=gh,h^2=i,b^a=bd,c^a=ce,d^a=dg,c^b=cg,d^b=dh,f^b=fh,h^b=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 45570);