Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #142801 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2021 |
| commutator, Frattini, cosocle | C2×#117 | C23 |
| prime radical | #18300 | C2 |
| LCS | G ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1 | |
624 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 624 conjugacy classes of subgroups — too many to draw. Only its 68 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) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,5,3)(4,6,8,7)(9,10,14,19)(11,20,23,33)(12,24,27,17)(13,16,28,18)(15,25,30,21)(22,35,38,36)(26,34,40,32)(29,37,31,39);(2,6)(3,7)(4,8)(9,11)(10,15)(12,25)(13,22)(14,23)(16,31)(17,33)(18,29)(19,30)(20,24)(21,27)(26,39)(28,38)(32,36)(34,35)(37,40);(1,3)(2,5)(4,6)(7,8)(10,16)(13,28)(15,29)(17,32)(18,19)(20,35)(21,25)(22,38)(24,34)(26,40)(30,31)(33,36)Polycyclic presentation:
2^9|a^2=g,d^2=g,e^2=i,f^2=gi,b^a=bd,c^a=ce,d^a=dg,e^a=ei,f^a=fhi,c^b=cf,d^b=dg,f^b=fgi,h^b=hi,d^c=dh,e^c=ei,f^c=fgi,f^d=fi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 50042);