Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2127 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C22×M16 | C23 |
| socle | C22 | #1459 |
| prime radical | C24 | C2×M16 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
414 conjugacy classes of subgroups; only the 142 normal subgroups are shown.
This group has 414 conjugacy classes of subgroups — too many to draw. Only its 142 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:
(1,2,4,3)(5,6,9,8,12,16,11,15)(7,17,19,29,21,18,23,13)(10,25,24,30,27,14,26,22)(20,31,33,36,35,32,34,28);(5,7,10,20,11,19,26,33,12,21,27,35,9,23,24,34)(6,13,14,28,15,17,30,31,16,29,25,36,8,18,22,32);(1,3,4,2)(5,8,9,16,12,15,11,6)(7,18,23,29,21,17,19,13)(10,22,24,25,27,30,26,14)(20,32,34,36,35,31,33,28)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=eg,d^2=i,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ci,f^a=fgi,d^b=dg,e^b=ei,g^b=gi,f^d=fi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35545);