Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500109 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#481 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×#469 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C23×M16 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1190 conjugacy classes of subgroups; only the 164 normal subgroups are shown.
This group has 1190 conjugacy classes of subgroups — too many to draw. Only its 164 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(6,10)(7,13)(8,16)(11,18)(12,15)(14,20)(17,19);(1,2)(3,4)(7,14)(10,18)(13,20)(15,19);(2,4)(5,6,8,17,9,11,16,12)(7,15,13,18,14,19,20,10);(7,14)(10,18)(13,20)(15,19)Polycyclic presentation:
2^9|c^2=eh,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=cg,f^b=fi,e^c=ei,f^c=fhi,h^c=hi,f^d=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72214893, #72224579, #72224582, … |
| maximal normal subgroup of | #72214893, #72224579, #72224582, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407305);