Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13156 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1936 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C2 | #9556 |
| 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 | |
242 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 242 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) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,6,9,7,13,5,23,8,14,4,20,21,12,22,11)(3,15,17,29,16,27,18,30,19,26,25,28,32,31,24,10);(2,9,12,11,14,20,13,23)(3,16,19,32)(4,6)(5,22)(10,27,29,26,30,31,28,15)(17,24,25,18);(1,3,4,17,7,16,22,18,8,19,6,25,21,32,5,24)(2,10,11,27,12,28,20,15,14,30,23,31,13,29,9,26)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=d,d^2=g,e^2=gi,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,g^a=gi,c^b=cg,d^b=di,e^b=ei,f^b=fi,e^c=ei,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72224949, #72224958, #72224959, … |
| maximal normal subgroup of | #72224949, #72224958, #72224959, … |
| maximal quotient of | C2×#140192 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47388);