Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #163012 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C25 | C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1122 conjugacy classes of subgroups; only the 218 normal subgroups are shown.
This group has 1122 conjugacy classes of subgroups — too many to draw. Only its 218 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) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10,13,18,14,17,11,15)(12,20,19,24,22,16,21,23)(25,26)(27,31)(28,36,37,40)(29,34)(30,33)(32,35,39,38);(1,3)(2,5)(4,8)(6,7)(9,11,14,13)(10,15,17,18)(12,21,22,19)(16,24,20,23)(25,27,30,29)(26,31,33,34)(28,35,37,38)(32,36,39,40);(3,8)(5,7)(9,12,14,22)(10,16,17,20)(11,19,13,21)(15,23,18,24)(25,28)(26,32)(27,35)(29,38)(30,37)(31,36)(33,39)(34,40);(2,6)(5,7)(26,33)(31,34)(32,39)(36,40)Polycyclic presentation:
2^9|a^2=g,b^2=hi,c^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dei,f^a=fh,c^b=ceh,f^c=fi,g^c=ghi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 120819);