Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13005 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1874 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C2 | #8939 |
| prime radical | #3067 | C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C8 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
409 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 409 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) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(2,8,11,23,13,24,12,18)(3,14,16,21)(4,17)(9,20)(10,26,28,27,29,32,30,22)(15,25,19,31);(1,2,9,8,5,12,4,18,7,13,20,24,6,11,17,23)(3,10,25,22,21,28,19,26,16,29,31,27,14,30,15,32);(1,3)(2,10)(4,19)(5,21)(6,14)(7,16)(8,22)(9,25)(11,30)(12,28)(13,29)(15,17)(18,26)(20,31)(23,32)(24,27)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,d^2=g,e^2=i,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=di,d^b=dg,e^b=ei,f^b=fg,g^b=gi,h^b=hi,d^c=di,e^c=ei,f^c=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72225439, #72225466, #72225469 |
| maximal normal subgroup of | #72225439, #72225466, #72225469 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43797);