Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33752 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C2×C4×D8 |
| prime radical | C23 | C2×#135 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
444 conjugacy classes of subgroups; only the 190 normal subgroups are shown.
This group has 444 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) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 48 points:
(1,2)(3,10)(4,8)(5,14)(6,9)(7,12)(11,16)(13,15)(17,18,21,27,23,28,22,37)(19,24,30,39,32,40,31,26)(20,33,34,38,36,47,35,48)(25,41,42,45,43,44,46,29);(1,3)(2,7)(4,13)(5,11)(6,12)(8,16)(9,10)(14,15)(17,19,21,30,23,32,22,31)(18,24,27,39,28,40,37,26)(20,29,34,41,36,45,35,44)(25,38,42,47,43,48,46,33);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,15,10,16)(17,20,31,44,22,35,32,45,23,36,30,41,21,34,19,29)(18,25,24,38,27,42,39,47,28,43,40,48,37,46,26,33);(1,5,6,4)(2,8,9,14)(3,11,12,13)(7,16,10,15)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=efh,d^2=e,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=ce104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 404298);