Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25443 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2366 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1002 conjugacy classes of subgroups; only the 234 normal subgroups are shown.
This group has 1002 conjugacy classes of subgroups — too many to draw. Only its 234 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:
(1,2)(3,8)(4,6)(5,7)(9,10,13,16,15,20,14,19)(11,21,22,17,24,30,29,32)(12,25,27,18,26,31,28,34)(23,37,36,35,39,33,38,40);(1,3,5,4)(2,6,7,8)(10,16,20,19)(11,22,24,29)(12,26)(17,32)(18,25,34,31)(21,30)(23,38,39,36)(27,28);(1,4,5,3)(2,6,7,8)(9,11,15,24)(10,17)(12,23,26,39)(13,22,14,29)(16,30)(18,33)(19,21)(20,32)(25,35)(27,36,28,38)(31,40)(34,37);(2,7)(6,8)(9,12,13,27,15,26,14,28)(10,18,19,25,20,34,16,31)(11,23,22,36,24,39,29,38)(17,33,21,35,32,37,30,40)Polycyclic presentation:
2^9|a^2=h,b^2=eg,c^2=eh,d^2=f,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,d^b=di,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 282620);