Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11499 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator | C24 | C22×C8 |
| Frattini, cosocle | C22×M16 | C23 |
| socle | C22 | C2×#324 |
| prime radical | C25 | C2×C8 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C23×M16 ◁ G | |
| LECS | G ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
700 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 700 conjugacy classes of subgroups — too many to draw. Only its 114 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 — 3 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10,12,17,13,18,23,31,14,15,22,26,24,27,35,37)(11,16,20,28,21,29,34,38,19,25,32,30,33,36,40,39);(2,5)(6,8)(10,15)(11,19)(12,22)(18,27)(21,33)(23,35)(28,30)(38,39);(1,3,4,7)(2,6,5,8)(9,11)(10,16,15,25)(12,20)(13,21)(14,19)(17,30,26,28)(18,29,27,36)(22,32)(23,34)(24,33)(31,39,37,38)(35,40)Polycyclic presentation:
2^9|a^2=f,c^2=d,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,g^a=gi,c^b=ci,f^b=fi,e^c=ei,f^c=fg,f^e=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72240699, #72240700, #72240701 |
| maximal normal subgroup of | #72240699, #72240700, #72240701 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 42380);