Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #133406 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1790 |
| commutator | C22×D4 | C22×C4 |
| Frattini, cosocle | C22×#41 | C23 |
| prime radical | #49562 | C2 |
| LCS | G ▷ C22×D4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C23 ▷ C2 ▷ C1 | |
931 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 931 conjugacy classes of subgroups — too many to draw. Only its 82 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)(9,10,13,16,14,17,11,18)(12,15,19,23,21,24,20,22);(1,3)(2,5)(4,8)(6,7)(10,15)(11,13)(16,23)(17,24)(18,22)(19,20);(9,11)(10,16)(12,20)(13,14)(15,22)(17,18)(19,21)(23,24)Polycyclic presentation:
2^9|a^2=f,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,h^a=hi,c^b=ci,e^b=ehi,f^b=fg,d^c=dh,f^c=fi,e^d=ei,f^d=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72226041, #72228003, #72232771, … |
| maximal normal subgroup of | #72226041, #72228003, #72232771, … |
| maximal quotient of | #72226032, #72226068, #72232769, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 40614);