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 | C23 | #450 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1770 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
296 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 296 conjugacy classes of subgroups — too many to draw. Only its 92 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,6)(3,7)(4,8)(9,10,13,19,15,20,14,30)(11,21,24,37,22,16,25,33)(12,26,27,34,29,17,28,18)(23,31,39,32,38,35,40,36);(1,2,5,3)(4,7,8,6)(9,11)(10,16,20,21)(12,23)(13,24)(14,25)(15,22)(17,31,26,35)(18,32,34,36)(19,33,30,37)(27,39)(28,40)(29,38);(1,3,5,2)(4,6,8,7)(10,17)(12,27)(13,14)(16,31)(18,30)(19,34)(20,26)(21,35)(23,39)(24,25)(28,29)(32,37)(33,36)(38,40)Polycyclic presentation:
2^9|a^2=f,b^2=fh,c^2=g,d^2=g,e^2=h,f^2=i,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 | #72309442, #72309446, #72309455, … |
| maximal normal subgroup of | #72309442, #72309446, #72309455, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43742);