Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12711 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1781 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C22×M16 | C23 |
| socle | C2 | #8298 |
| prime radical | C23×#41 | C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×#568 ◁ G | |
| LECS | G ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
1414 conjugacy classes of subgroups; only the 132 normal subgroups are shown.
This group has 1414 conjugacy classes of subgroups — too many to draw. Only its 132 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,11,6,12,21,29,7,8,20,23,4,10,18,27)(3,13,15,30,14,22,31,28,16,26,19,32,17,9,24,25);(2,8)(9,22)(10,12)(13,26);(1,3)(2,9)(4,17)(5,19)(6,14)(7,16)(8,22)(10,26)(11,28)(12,13)(15,20)(18,31)(21,24)(23,25)(27,30)(29,32)Polycyclic presentation:
2^9|a^2=f,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,f^b=fg,e^c=ei,f^c=fi,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72292724, C2×#129238 |
| maximal normal subgroup of | #72292724, C2×#129238 |
| maximal quotient of | C2×#129238 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36434);