Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #144372 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2021 |
| commutator, Frattini, cosocle | C23×Q8 | C23 |
| prime radical | #3486 | C22 |
| LCS | G ▷ C23×Q8 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×Q8 ◁ G | |
| LECS | G ▷ C23×Q8 ▷ C23 ▷ C2 ▷ C1 | |
909 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 909 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) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,19)(12,18)(13,23)(14,24)(15,20)(16,22)(17,21);(5,8)(9,11,15,14)(10,16,20,17)(12,22,13,21)(18,24,23,19);(1,3)(2,5)(4,7)(6,8)(9,12,15,13)(10,17,20,16)(11,21,14,22)(18,19,23,24)Polycyclic presentation:
2^9|b^2=i,c^2=i,d^2=i,e^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fghi,g^a=gi,h^a=hi,c^b=cf,d^b=di,e^b=eh,d^c=dg,e^c=ei,e^d=ei44 conjugacy classes — view the full character table.
| maximal subgroup of | #72267806, #72267850, #72268115, … |
| maximal normal subgroup of | #72267806, #72267850, #72268115, … |
| maximal quotient of | #72267809, #72267819, #72267859 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 51570);