Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499946 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #3109 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #13331 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
623 conjugacy classes of subgroups; only the 122 normal subgroups are shown.
This group has 623 conjugacy classes of subgroups — too many to draw. Only its 122 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 — 4 generators on 24 points:
(10,15)(11,19)(12,13)(16,20)(17,24)(18,23)(21,22);(1,2)(3,6)(4,5)(7,8)(10,16,17,18)(15,23,24,20);(2,5)(6,8)(9,10)(11,20)(12,18)(13,16)(14,17)(15,19)(21,23)(22,24);(1,3,4,7)(2,6,5,8)(11,21)(15,24)(19,22)(20,23)Polycyclic presentation:
2^9|b^2=e,d^2=gh,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,h^a=hi,c^b=cg,e^c=ei,f^c=fh,g^c=gi,h^c=hi,f^d=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72265494, #72265585, #72265615 |
| maximal normal subgroup of | #72265494, #72265585, #72265615 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 414928);