Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #140281 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1936 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #3169 | C22 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1451 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
227 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 227 conjugacy classes of subgroups — too many to draw. Only its 66 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) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,4,6,5,9,13,8)(3,10,12,16,11,7,15,14)(17,18,23,27,24,28,31,19)(20,22,30,25,26,21,29,32);(2,6,9,8)(3,11)(7,16,10,14)(12,15)(17,19,21,30,23,28,32,20,24,27,22,29,31,18,25,26);(1,3,4,12,5,11,13,15)(2,7,8,16,9,10,6,14)(17,20,21,19,23,29,32,28,24,26,22,27,31,30,25,18)Polycyclic presentation:
2^9|a^2=f,b^2=dfgi,c^2=d,d^2=g,e^2=h,f^2=gh,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=cg,d^b=di,e^b=ei,f^b=fi,e^c=eghi,f^c=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72227061, #72227161, #72227177, … |
| maximal normal subgroup of | #72227061, #72227161, #72227177, … |
| maximal quotient of | #72227054 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47528);