Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7035 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #95 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C23 | C2×#95 |
| prime radical | C26 | Q8 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C25×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C23 ▷ C1 | |
1966 conjugacy classes of subgroups; only the 198 normal subgroups are shown.
This group has 1966 conjugacy classes of subgroups — too many to draw. Only its 198 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) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,15,16,13)(12,14)(17,18,20,24,22,26,21,25)(19,27,28,23,30,31,29,32);(1,3)(2,5,6,7)(4,8)(9,11)(10,13)(12,16)(14,15)(17,19,21,29,22,30,20,28)(18,23,24,31,26,32,25,27);(17,20,22,21)(18,24,26,25)(19,28,30,29)(23,31,32,27)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=gi,d^2=gi,e^2=gi,f^2=gi,b^a=bd,d^a=dg,f^a=fghi,d^b=dh,e^b=ei104 conjugacy classes — view the full character table.
| Fitting subgroup of | #72266635 |
| maximal subgroup of | #72266589, #72266635, #72267823, … |
| maximal normal subgroup of | #72266589, #72266635, #72267823, … |
| maximal quotient of | #72266598, #72268048 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2529);