Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #480690 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×Q8 ◁ C22×2−1+4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1668 conjugacy classes of subgroups; only the 206 normal subgroups are shown.
This group has 1668 conjugacy classes of subgroups — too many to draw. Only its 206 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)(3,7)(4,6)(5,8)(9,10)(11,16)(12,23)(13,18)(14,19,28,29)(15,20)(17,25,32,22)(21,30)(24,33)(26,35,39,40)(27,36)(31,38,34,37);(1,3)(2,5)(4,8)(6,7)(9,11,15,12)(10,16,20,23)(13,24,27,21)(14,22,28,25)(17,29,32,19)(18,33,36,30)(26,38,39,37)(31,35,34,40);(3,8)(5,7)(9,12,15,11)(10,17,20,32)(13,21,27,24)(14,22,28,25)(16,29,23,19)(18,34,36,31)(26,38,39,37)(30,35,33,40);(2,6)(5,7)(9,13,15,27)(10,18,20,36)(11,21,12,24)(14,26,28,39)(16,30,23,33)(17,31,32,34)(19,35,29,40)(22,37,25,38)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=i,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=ce,d^b=di,d^c=di,f^c=fi,g^c=ghi,g^f=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403402);