Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #143326 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1634 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C2 | #10235 |
| prime radical | #28249 | C2 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
384 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 384 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) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,8,14)(3,15)(4,20)(5,25,6,29)(7,12,27,13)(9,32)(10,31,11,28)(16,24)(17,22)(18,21)(19,26)(23,30);(1,3)(2,4)(5,17)(6,30)(7,9)(8,19)(10,21)(11,32)(12,23)(13,22)(14,24)(15,29)(16,31)(18,27)(20,28)(25,26);(1,4)(2,9)(3,13)(5,26)(6,15)(7,23)(8,24)(10,30)(11,17)(12,19)(14,18)(16,29)(20,25)(21,28)(22,27)(31,32)Polycyclic presentation:
2^9|a^2=gh,d^2=g,e^2=h,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,c^b=cf,d^b=dgi,e^b=ei,f^b=fi,g^b=gi,d^c=di,e^c=ehi,f^c=fi,h^c=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 50992);