Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #144607 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2029 |
| commutator, Frattini, cosocle | C22×C4⋊C4 | C23 |
| prime radical | #3492 | C22 |
| LCS | G ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
487 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 487 conjugacy classes of subgroups — too many to draw. Only its 52 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:
(2,6)(3,13,17,19)(9,25,27,32)(10,24)(11,12)(14,31,15,30)(16,28,23,26)(20,29);(1,2,7,11)(3,14)(4,6,8,12)(5,10,21,20)(9,23)(13,19)(15,17)(16,27)(18,24,22,29)(25,32)(26,28)(30,31);(1,3)(2,9)(4,14)(5,19)(6,23)(7,17)(8,15)(10,28)(11,27)(12,16)(13,21)(18,30)(20,26)(22,31)(24,32)(25,29)Polycyclic presentation:
2^9|a^2=dgi,b^2=di,e^2=g,f^2=g,b^a=bd,c^a=ce,f^a=fghi,c^b=cf,e^b=eh,d^c=dg,e^c=eg,f^c=fg,h^c=hi,f^e=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72233826, #72233834, #72233835 |
| maximal normal subgroup of | #72233826, #72233834, #72233835 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 51822);