Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #144627 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2029 |
| commutator, Frattini, cosocle | C23×D4 | C23 |
| prime radical | #3484 | C22 |
| LCS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
663 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 663 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:
(1,2,7,21)(3,12,15,29)(4,16,17,10)(5,9,6,11)(8,22,24,14)(13,23,18,26)(19,25,20,27)(28,31,32,30);(3,13,14,30)(4,17)(8,23,12,28)(10,16)(15,18,22,31)(19,20)(24,26,29,32)(25,27);(1,3)(2,8)(4,18)(5,14)(6,15)(7,22)(9,12)(10,26)(11,24)(13,19)(16,28)(17,30)(20,31)(21,29)(23,27)(25,32)Polycyclic presentation:
2^9|a^2=dgi,b^2=d,f^2=i,b^a=bd,c^a=ce,e^a=eg,f^a=fghi,c^b=cf,e^b=eh,f^b=fgi,d^c=dg,f^c=fi,h^c=hi,f^e=fi38 conjugacy classes — view the full character table.
| maximal subgroup of | #72236957, #72236961, #72236965, … |
| maximal normal subgroup of | #72236957, #72236961, #72236965, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 51840);