Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #136516 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1864 |
| 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 | |
277 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 277 conjugacy classes of subgroups — too many to draw. Only its 76 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) = 256 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,6,9,7,4,5,10)(3,11,13,8,15,16,14,12)(17,18,21,25,22,27,19,23)(20,29,31,32,30,24,28,26);(1,3,5,14,7,15,6,13)(2,8,10,11,4,12,9,16)(18,23,27,25)(20,30)(24,26,29,32)(28,31);(1,4)(2,7)(3,12)(5,9)(6,10)(8,15)(11,13)(14,16)(18,24)(19,21)(20,31)(23,26)(25,32)(27,29)(28,30)Polycyclic presentation:
2^9|a^2=f,b^2=fh,d^2=g,e^2=h,f^2=g,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,d^b=dgi,e^b=eg,d^c=dg,e^c=ehi,f^c=fi,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72225273, #72227061, #72227172, … |
| maximal normal subgroup of | #72225273, #72227061, #72227172, … |
| maximal quotient of | #72225256 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43763);