Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #103197 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #392 |
| commutator, Frattini, cosocle | C24×C4 | C23 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C24×C4 ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C23 ▷ C1 | |
627 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 627 conjugacy classes of subgroups — too many to draw. Only its 80 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 — 3 generators on 40 points:
(2,6)(3,7)(4,8)(9,10,14,18)(11,19,21,24)(12,17,13,16)(15,22,23,20)(25,26,29,32)(27,30,34,37)(28,31,35,39)(33,36,40,38);(1,2)(3,5)(4,7)(6,8)(9,11,14,21)(10,15,18,23)(12,22,13,20)(16,24,17,19)(25,27,29,34)(26,30,32,37)(28,33,35,40)(31,38,39,36);(1,3,5,2)(4,6,8,7)(9,11)(10,15)(12,20)(13,22)(14,21)(16,19)(17,24)(18,23)(25,28)(26,31)(27,33)(29,35)(30,36)(32,39)(34,40)(37,38)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=i,d^2=g,e^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fhi,c^b=cf,d^b=dg,e^b=ei,d^c=dh,e^c=eg50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10431);