Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #145015 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2041 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #28467 | 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 | |
426 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 426 conjugacy classes of subgroups — too many to draw. Only its 78 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)(3,5)(4,6)(7,8)(9,10)(11,20)(12,24)(13,29)(14,19)(15,21)(16,25)(17,35)(18,27)(22,31)(23,34)(26,33)(28,36)(30,37)(32,39)(38,40);(2,5)(6,7)(11,21)(12,25)(13,27)(15,31)(16,24)(17,36)(18,29)(20,32)(22,39)(23,26)(28,35)(30,40)(33,37)(34,38);(2,6)(4,8)(5,7)(9,11,13,30,14,22,27,23)(10,15,18,38,19,32,29,33)(12,26,25,39,28,40,35,21)(16,34,24,31,17,37,36,20)Polycyclic presentation:
2^9|c^2=dg,e^2=gi,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eg,f^a=fhi,g^a=gi,h^a=hi,c^b=cf,e^b=eh,f^b=fh,g^b=gi,h^b=hi,d^c=di,e^c=egi,f^c=fhi,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, 52225);