Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #482975 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#102 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C23 | C22×SD16 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×Q8 ◁ C2×C8×Q8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
655 conjugacy classes of subgroups; only the 236 normal subgroups are shown.
This group has 655 conjugacy classes of subgroups — too many to draw. Only its 236 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) = 256 |
A minimal faithful permutation representation — 4 generators on 48 points:
(2,7,9,13)(8,14,16,15)(17,18,21,26,23,28,22,27)(19,29,31,42,33,46,32,45)(20,34,36,25,38,43,37,44)(24,35,39,30,41,47,40,48);(1,2)(3,8)(4,7)(5,13)(6,9)(10,15)(11,16)(12,14)(17,19,23,33)(18,24,28,41)(20,35,38,47)(21,32,22,31)(25,42,44,45)(26,40,27,39)(29,34,46,43)(30,36,48,37);(1,3,6,11)(2,8,9,16)(4,12,5,10)(7,14,13,15);(1,4,6,5)(2,7,9,13)(3,10,11,12)(8,15,16,14)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=gh,d^2=gh,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=be,e^a=eg,g^a=gi,h^a=hi,e^b=eh,f^b=fghi,g^b=gi,h^b=hi,d^c=dgh77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 390200);