Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #488889 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#447 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| prime radical | #3147 | C22 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C8×Q8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
712 conjugacy classes of subgroups; only the 156 normal subgroups are shown.
This group has 712 conjugacy classes of subgroups — too many to draw. Only its 156 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) = 256 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,5)(3,7)(6,8)(9,10)(11,20)(12,15)(13,28)(14,17)(16,21)(18,24)(19,26)(22,39)(23,36)(25,35)(27,30)(29,34)(31,37)(32,38)(33,40);(1,2,4,6)(3,8,7,5)(10,15)(12,24)(13,26)(16,20)(17,30)(18,19)(21,25)(22,29)(27,37)(28,31)(32,36)(33,39)(34,35)(38,40);(9,11,14,23)(10,16,17,32)(12,25,27,40)(13,29,26,22)(15,20,30,36)(18,34,31,39)(19,35,28,33)(21,37,38,24);(10,17)(15,30)(16,32)(18,31)(19,28)(20,36)(33,35)(34,39)Polycyclic presentation:
2^9|b^2=ghi,c^2=i,e^2=gi,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,e^a=eg,f^a=fh,g^a=gi,h^a=hi,e^b=eg,f^b=fh,g^b=gi,h^b=hi,e^c=eh,f^c=fh,g^c=gi,h^c=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 396234);