Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #143235 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2011 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #33719 | C2 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1770 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
514 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 514 conjugacy classes of subgroups — too many to draw. Only its 82 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:
(1,2)(3,5)(4,7)(6,8)(9,10)(11,25)(12,30)(13,21)(14,20)(15,24)(16,23)(17,22)(18,26)(19,32)(27,36)(28,35)(29,38)(31,37)(33,39)(34,40);(2,5)(3,8)(6,7)(9,11)(10,18)(12,15)(13,32)(14,26)(16,29)(17,28)(19,22)(20,38)(21,25)(23,36)(24,37)(27,33)(30,39)(31,34)(35,40);(9,12,15,11)(10,19,24,36)(13,32,33,27)(14,26,34,31)(16,28,17,29)(18,22,37,23)(20,30,40,25)(21,38,39,35)Polycyclic presentation:
2^9|c^2=i,d^2=gi,e^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fi,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=ehi,g^b=gi,h^b=hi,d^c=dh,e^c=eh,g^c=gi,h^c=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 50432);