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 | C23×D4 | C23 |
| prime radical | #47846 | C2 |
| LCS | G ▷ C22×D4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
976 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 976 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)(11,18,20,15)(12,16,22,23)(13,17)(14,19)(21,24);(1,3)(2,5)(4,8)(6,7)(9,11,13,20)(10,14,17,24)(12,19,22,21)(15,16,18,23);(9,11)(10,15)(12,21)(13,20)(14,23)(16,24)(17,18)(19,22)Polycyclic presentation:
2^9|a^2=f,b^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 | #72232811, #72233357, #72233373, … |
| maximal normal subgroup of | #72232811, #72233357, #72233373, … |
| maximal quotient of | #72232807, #72233406, #72233633, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 40608);