Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #346294 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C26 | D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1856 conjugacy classes of subgroups; only the 190 normal subgroups are shown.
This group has 1856 conjugacy classes of subgroups — too many to draw. Only its 190 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:
(1,2)(3,6)(4,5)(7,8)(9,10,11,13)(12,14,16,15)(21,24);(2,5)(6,8)(10,13)(14,15);(1,3,4,7)(2,6,5,8)(10,14)(12,16)(13,15)(17,18)(19,21)(20,22)(23,24);(17,19)(18,21)(20,23)(22,24)Polycyclic presentation:
2^9|a^2=h,c^2=eh,f^2=h,b^a=be,c^a=cf,d^a=dg,f^a=fh,f^c=fh,g^c=gi,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72209853, #72227682 |
| maximal normal subgroup of | #72209853, #72227682 |
| maximal quotient of | #72227776 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 257008);