Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6433040 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1558 conjugacy classes of subgroups; only the 554 normal subgroups are shown.
This group has 1558 conjugacy classes of subgroups — too many to draw. Only its 554 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) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,5)(3,6,7,8)(9,10)(11,14)(12,19)(13,15)(16,20)(17,22)(18,24)(21,23)(25,26,30,35)(27,36,39,32)(28,31,38,40)(29,33,37,34);(2,5)(6,8)(9,11,13,17)(10,14,15,22)(12,18,20,23)(16,21,19,24)(25,27,37,38,30,39,29,28)(26,31,33,32,35,40,34,36);(1,3,4,7)(2,6,5,8)(10,15)(11,17)(16,19)(18,23)(25,28,29,39,30,38,37,27)(26,32,34,31,35,36,33,40);(9,12,13,20)(10,16,15,19)(11,18,17,23)(14,21,22,24);(25,29,30,37)(26,33,35,34)(27,38,39,28)(31,36,40,32)Polycyclic presentation:
2^9|a^2=fgh,b^2=gi,c^2=f,d^2=h,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,g^a=gi,c^b=ch,e^b=ei,e^c=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6343545);