Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501120 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×Q8 | C24 |
| prime radical | C23 | C2×#120 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
723 conjugacy classes of subgroups; only the 150 normal subgroups are shown.
This group has 723 conjugacy classes of subgroups — too many to draw. Only its 150 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 — 4 generators on 40 points:
(1,2,5,8)(3,7,4,6)(9,10,13,16)(11,21,24,32)(12,19,14,18)(15,29,27,20)(17,23,30,25)(22,37,39,36)(26,34,28,35)(31,40,33,38);(1,3,5,4)(2,6,8,7)(10,16)(11,22)(15,27)(17,31)(18,19)(21,36)(23,38)(24,39)(25,40)(26,28)(30,33)(32,37);(9,11,14,25,13,24,12,23)(10,17,19,21,16,30,18,32)(15,22,26,38,27,39,28,40)(20,33,34,36,29,31,35,37);(1,4,5,3)(2,7,8,6)(9,12,13,14)(10,18,16,19)(11,23,24,25)(15,26,27,28)(17,32,30,21)(20,34,29,35)(22,38,39,40)(31,37,33,36)Polycyclic presentation:
2^9|a^2=e,b^2=ei,c^2=f,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=ghi,c^b=cg,f^b=fh,g^b=gi,g^c=ghi,h^c=hi,g^f=gi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 410558);