Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #133434 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1782 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C23×Q8 | C23 |
| prime radical | #2815 | C4 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ #2310 ◁ G | |
| LECS | G ▷ C23×Q8 ▷ C23 ▷ C2 ▷ C1 | |
646 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 646 conjugacy classes of subgroups — too many to draw. Only its 78 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 — 3 generators on 40 points:
(5,8)(9,10,13,18,14,19,26,34)(11,20,23,35,22,16,30,32)(12,15,25,29,24,17,27,33)(21,31,37,40,36,28,38,39);(1,2)(3,5)(4,6)(7,8)(10,15)(12,24)(13,26)(16,28)(17,19)(18,29)(20,31)(21,36)(23,30)(32,39)(33,34)(35,40);(1,3)(2,5)(4,7)(6,8)(9,11)(10,16,19,20)(12,21)(13,23)(14,22)(15,28,17,31)(18,32,34,35)(24,36)(25,37)(26,30)(27,38)(29,39,33,40)Polycyclic presentation:
2^9|a^2=f,c^2=g,d^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=di,e^b=eh,f^b=fgi,d^c=dh,e^c=ei,f^d=fi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 40633);