Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #144386 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2021 |
| 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 | |
868 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 868 conjugacy classes of subgroups — too many to draw. Only its 60 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,5)(3,7)(6,8)(9,10,13,16)(11,14,18,21)(12,15,19,23)(17,20,24,22);(3,7)(9,11,13,18)(10,14,16,21)(12,17,19,24)(15,22,23,20);(1,2)(3,8)(4,6)(5,7)(9,12)(10,15)(11,17)(13,19)(14,20)(16,23)(18,24)(21,22)Polycyclic presentation:
2^9|a^2=g,b^2=g,e^2=i,b^a=bd,c^a=ce,e^a=ei,f^a=fghi,c^b=cf,e^b=eh,d^c=dg,e^c=ei,h^c=hi,f^e=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72236527, #72236679, #72236692, … |
| maximal normal subgroup of | #72236527, #72236679, #72236692, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 51624);