Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476707 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2826 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | 2−1+4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
526 conjugacy classes of subgroups; only the 154 normal subgroups are shown.
This group has 526 conjugacy classes of subgroups — too many to draw. Only its 154 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)(3,6)(4,5)(7,8)(9,10,15,20,16,22,14,21)(11,23,26,37,27,38,25,17)(12,28,30,18,31,39,34,36)(13,32,24,40,29,19,33,35);(2,5)(3,7)(9,11,16,27)(10,17)(12,29)(13,31)(14,25,15,26)(18,32,36,19)(20,23)(21,38)(22,37)(24,34)(28,35,39,40)(30,33);(3,7)(6,8)(9,12,15,30,16,31,14,34)(10,18,21,28,22,36,20,39)(11,24,25,13,27,33,26,29)(17,35,23,32,37,40,38,19);(1,3,4,7)(2,6,5,8)(9,13,14,24,16,29,15,33)(10,19,20,40,22,32,21,35)(11,12,25,30,27,31,26,34)(17,36,23,39,37,18,38,28)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=f,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=def,f^b=fi,d^c=de,e^c=ei,h^c=hi,g^d=gi,h^d=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 386977);