Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500599 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #3109 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×D8 | C23 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
527 conjugacy classes of subgroups; only the 144 normal subgroups are shown.
This group has 527 conjugacy classes of subgroups — too many to draw. Only its 144 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(9,10)(11,20)(12,26)(13,21)(14,31)(15,19)(16,22)(17,24)(18,23)(25,34)(27,37)(28,35)(29,40)(30,33)(32,36)(38,39);(1,2,4,6)(3,7,8,5)(9,11,15,14,16,25,13,24)(10,17,21,34,22,31,19,20)(12,27,28,39,30,32,29,23)(18,35,36,26,38,40,37,33);(1,3,4,8)(2,5,6,7)(9,12)(10,18)(11,23)(13,28)(14,32)(15,29)(16,30)(17,33)(19,36)(20,35)(21,37)(22,38)(24,27)(25,39)(26,31)(34,40);(9,11,13,24,16,25,15,14)(10,17,19,20,22,31,21,34)(12,23,28,27,30,39,29,32)(18,33,36,35,38,26,37,40)Polycyclic presentation:
2^9|b^2=g,c^2=gh,d^2=e,e^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,g^a=gi,h^a=hi,c^b=cg,d^b=di,f^b=fi,f^c=fh,g^c=gi,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72265538, #72265596, #72265615, … |
| maximal normal subgroup of | #72265538, #72265596, #72265615, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 414775);