Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12952 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1864 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C2 | #8845 |
| prime radical | C2×#135 | C2×C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C8 ◁ C4×M32 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
294 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 294 conjugacy classes of subgroups — too many to draw. Only its 102 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:
(1,2,5,12,7,13,16,10,8,14,6,24,26,4,23,11)(3,15,19,17,21,30,32,9,22,27,25,28,20,31,18,29);(1,3,16,18,26,20,5,25,8,22,23,32,7,21,6,19)(2,9,11,31,4,17,24,27,14,29,10,30,13,28,12,15);(1,4)(2,7)(3,17)(5,10)(6,11)(8,13)(9,21)(12,23)(14,26)(15,32)(16,24)(18,27)(19,31)(20,29)(22,28)(25,30)Polycyclic presentation:
2^9|a^2=f,b^2=fg,d^2=g,e^2=g,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,d^b=dg,e^b=egi,f^b=fi,g^b=gi,d^c=dgi,e^c=egi,f^c=fi,g^c=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72225460, #72225466, #72225469 |
| maximal normal subgroup of | #72225460, #72225466, #72225469 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43108);