Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #133338 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1790 |
| commutator | C22×Q8 | C22×C4 |
| Frattini, cosocle | C23×Q8 | C23 |
| prime radical | #45977 | C2 |
| LCS | G ▷ C22×Q8 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×Q8 ◁ G | |
| LECS | G ▷ C23×Q8 ▷ C23 ▷ C2 ▷ C1 | |
973 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 973 conjugacy classes of subgroups — too many to draw. Only its 84 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)(3,7,8,5)(4,6)(9,10,13,16,14,19,12,18)(11,15,22,23,20,17,21,24);(1,3)(2,5)(4,8)(6,7)(10,15)(11,20)(12,13)(16,23)(17,19)(18,24);(10,16)(12,13)(15,23)(17,24)(18,19)(21,22)Polycyclic presentation:
2^9|a^2=f,d^2=i,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,h^a=hi,d^b=di,e^b=ehi,f^b=fgi,d^c=dh,e^c=ei,f^c=fi,e^d=ei,f^d=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72238625 |
| maximal normal subgroup of | #72238625 |
| maximal quotient of | #72238634, #72238635, #72238680, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 40540);