Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12416 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | #459 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
390 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 390 conjugacy classes of subgroups — too many to draw. Only its 114 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 44 points:
(1,2)(3,4)(6,9)(7,10)(11,12)(13,14,17,23,33,42,31,20,19,25,16,22,18,24,34,38)(15,26,28,44,36,41,32,40,30,37,27,39,29,43,35,21);(5,6,10,12,8,11,7,9)(14,20)(16,31)(17,34)(18,33)(21,37)(22,24)(23,42)(25,38)(26,40)(27,32)(28,35)(29,36)(39,43)(41,44);(2,4)(13,15,19,30)(14,21,25,40)(16,32,17,35)(18,36,33,29)(20,37,38,26)(22,41,23,43)(24,44,42,39)(27,34,28,31)Polycyclic presentation:
2^9|a^2=f,b^2=dfghi,c^2=i,d^2=ghi,e^2=hi,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,d^b=dh,e^b=eh,f^b=fh,h^b=hi,d^c=dh,e^c=eh,f^c=fh,h^c=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 45824);