Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #153078 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×D16 |
| commutator, Frattini, cosocle | C22×C16 | C23 |
| prime radical | C22×D16 | C22 |
| LCS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
391 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 391 conjugacy classes of subgroups — too many to draw. Only its 88 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,4)(10,16)(11,19)(12,22)(13,25)(14,30)(17,34)(18,36)(20,37)(21,33)(23,28)(24,29)(26,40)(27,32)(31,35)(38,39);(2,4)(5,6)(7,8)(9,10,15,11)(12,23,27,24)(13,28,31,29)(14,21,30,18)(16,32,19,22)(17,35,20,25)(26,37,38,34)(33,39,36,40);(6,8)(9,11,15,10)(12,24,27,23)(13,29,31,28)(14,18,30,21)(16,22,19,32)(17,25,20,35)(26,34,38,37)(33,40,36,39)Polycyclic presentation:
2^9|b^2=i,c^2=i,d^2=gh,e^2=gh,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gh,h^a=hi,c^b=cf,d^b=dg,e^b=eg,g^b=gh,h^b=hi,d^c=dg,e^c=eg,g^c=gh,h^c=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60297);