Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11669 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #418 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1490 |
| prime radical | C24 | #104 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
481 conjugacy classes of subgroups; only the 154 normal subgroups are shown.
This group has 481 conjugacy classes of subgroups — too many to draw. Only its 154 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,9,14,10,16,25,31,11,17,26,13,24,15,8,23)(7,18,20,28,21,33,35,29,22,30,36,27,34,32,19,12);(2,4)(5,7,10,21,11,22,24,34)(6,12)(8,19,9,20,25,35,26,36)(13,30)(14,18)(15,27)(16,28)(17,29)(23,32)(31,33);(5,8,10,9,11,25,24,26)(6,13,15,31,17,14,16,23)(7,19,21,20,22,35,34,36)(12,18,27,32,29,30,28,33)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=g,d^2=gi,e^2=h,f^2=gh,g^2=i,h^2=i,b^a=bd,c^a=ch,d^a=dg,f^a=fgi,g^a=gi,c^b=cghi,d^c=di,e^c=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34091);