Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12316 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #447 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1879 |
| prime radical | C23 | #463 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C8×M16 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
301 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 301 conjugacy classes of subgroups — too many to draw. Only its 78 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 — 3 generators on 48 points:
(2,6,8,11)(3,9,10,12)(17,18,23,29,21,27,24,38)(19,28,22,36,31,25,32,41)(20,33,35,40,34,48,39,45)(26,37,43,46,42,47,44,30);(1,2,4,11,5,8,13,6)(3,7,12,15,10,16,9,14)(17,19,24,32,21,31,23,22)(18,25,38,36,27,28,29,41)(20,30,39,47,34,46,35,37)(26,40,44,33,42,45,43,48);(1,3)(2,7)(4,12)(5,10)(6,14)(8,16)(9,13)(11,15)(17,20,21,34)(18,26,27,42)(19,30,31,46)(22,37,32,47)(23,35,24,39)(25,40,28,45)(29,43,38,44)(33,41,48,36)Polycyclic presentation:
2^9|a^2=f,b^2=df,c^2=i,d^2=g,e^2=ghi,f^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eh,h^a=hi,d^b=dg,e^b=eg,f^b=fg,d^c=dg,e^c=egh,f^c=fg,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44682);