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 | |
649 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 649 conjugacy classes of subgroups — too many to draw. Only its 84 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)(3,4)(7,8)(9,10,13,16)(11,14,18,21)(12,15,19,23)(17,20,24,22)(25,26)(27,30)(28,31)(29,32)(33,38)(34,37)(35,39)(36,40);(1,2)(3,8)(4,6)(5,7)(9,11,13,18)(10,14,16,21)(12,17,19,24)(15,22,23,20)(25,27)(26,30)(28,33)(29,34)(31,38)(32,37)(35,40)(36,39);(1,2)(3,6)(4,8)(5,7)(9,12)(10,15)(11,17)(13,19)(14,20)(16,23)(18,24)(21,22)(25,28,29,35)(26,31,32,39)(27,33,34,40)(30,36,37,38)Polycyclic presentation:
2^9|a^2=h,b^2=h,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, 10420);