Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499855 |
| 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 | C22×#41 | C23 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
1007 conjugacy classes of subgroups; only the 134 normal subgroups are shown.
This group has 1007 conjugacy classes of subgroups — too many to draw. Only its 134 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 — 4 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10,13,15)(11,19,21,16)(12,23,25,35)(14,28,27,18)(17,30,31,26)(20,36,37,40)(22,33,29,34)(24,39,32,38);(2,5)(6,8)(9,11,13,21)(10,16,15,19)(12,24,25,32)(14,29,27,22)(17,20,31,37)(18,34,28,33)(23,38,35,39)(26,36,30,40);(1,3,4,7)(2,6,5,8)(9,12,13,25)(10,17,15,31)(11,20,21,37)(14,30,27,26)(16,32,19,24)(18,35,28,23)(22,39,29,38)(33,36,34,40);(2,5)(6,8)(9,10,13,15)(11,19,21,16)(12,17,25,31)(14,18,27,28)(20,24,37,32)(22,34,29,33)(23,30,35,26)(36,39,40,38)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=e,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dei,f^a=fi,g^a=ghi,c^b=cg,d^b=di,f^b=fh,g^b=gi,f^c=fi,g^c=gi,h^c=hi,g^f=gi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407413);