Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #133912 |
| 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 | C22×#41 | C2×C4 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ #2310 ◁ G | |
| LECS | G ▷ C23×Q8 ▷ C23 ▷ C2 ▷ C1 | |
877 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 877 conjugacy classes of subgroups — too many to draw. Only its 86 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:
(1,2)(3,7,8,5)(4,6)(9,10,15,20,16,21,12,26)(11,22,23,38,25,34,31,33)(13,29,27,18,14,19,28,37)(17,32,35,30,36,39,40,24);(1,3,4,8)(2,5,6,7)(9,11)(10,17,21,36)(12,23)(13,24)(14,30)(15,31)(16,25)(18,33,37,38)(19,22,29,34)(20,35,26,40)(27,39)(28,32);(9,12,16,15)(10,18,21,37)(11,23,25,31)(13,28,14,27)(17,33,36,38)(19,20,29,26)(22,35,34,40)(24,32,30,39)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=i,d^2=i,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,e^b=ei,f^b=fgi,d^c=di,e^c=ei,f^c=fhi,f^d=fi,f^e=fi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 41111);