Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11669 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #418 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C2 | #8204 |
| prime radical | C2×#135 | C2×C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ C4×M32 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
309 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 309 conjugacy classes of subgroups — too many to draw. Only its 112 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:
(1,2,5,13,8,15,24,32,9,16,25,14,7,26,6,12)(3,17,4,11,21,28,23,31,22,30,18,10,20,27,19,29);(1,3,6,19,7,20,25,18,9,22,24,23,8,21,5,4)(2,10,14,28,15,29,12,30,16,11,13,27,26,31,32,17);(1,4,5,21,8,23,24,22,9,18,25,20,7,19,6,3)(2,11,13,28,15,31,32,30,16,10,14,27,26,29,12,17)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=e,d^2=gi,e^2=h,f^2=g,g^2=i,h^2=i,b^a=bd,d^a=dg,f^a=fg,g^a=gi,c^b=cgh,d^b=di,d^c=di92 conjugacy classes — view the full character table.
| maximal subgroup of | #72214092, #72214217, #72214250, … |
| maximal normal subgroup of | #72214092, #72214217, #72214250, … |
| maximal quotient of | #72216233, C3×#128726, #72226296, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35922);