Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #133834 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1781 |
| commutator | C25 | C22×C4 |
| Frattini, cosocle | C22×C22⋊C4 | C23 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C25 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#120 ◁ G | |
| LECS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1180 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 1180 conjugacy classes of subgroups — too many to draw. Only its 82 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:
(1,2,5,8,6,10,3,12)(4,15,17,21,18,24,13,28)(7,11,19,22,20,27,14,29)(9,23,25,31,26,30,32,16);(1,3)(2,8)(4,13)(5,6)(7,14)(9,21)(10,12)(11,22)(15,25)(16,30)(17,18)(19,20)(23,31)(24,32)(26,28)(27,29);(1,4,6,18)(2,9,10,26)(3,13,5,17)(7,16,20,31)(8,21,12,28)(11,15,27,24)(14,30,19,23)(22,25,29,32)Polycyclic presentation:
2^9|a^2=f,c^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,e^b=ei,f^b=fg,d^c=di,f^c=fh,f^e=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72282933, #72282959, #72283016, … |
| maximal normal subgroup of | #72282933, #72282959, #72283016, … |
| maximal quotient of | #72284123 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 41263);