Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2081 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #397 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C22 | #1439 |
| prime radical | C24 | C4×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
717 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,3,5,4,6,7,8)(9,10)(11,20)(12,25,29,17)(13,30)(14,18)(15,19)(16,22)(21,36,39,33)(23,37)(24,32)(26,31,38,40)(27,34,28,35);(9,11,13,22,15,23,14,24)(10,16,18,20,19,32,30,37)(12,26,27,21,29,38,28,39)(17,33,34,40,25,36,35,31);(9,12,15,29)(10,17,19,25)(11,21,23,39)(13,28,14,27)(16,31,32,40)(18,35,30,34)(20,36,37,33)(22,26,24,38)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=h,d^2=h,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=cf,e^b=eh,d^c=dh,f^c=fh,g^c=gh92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 24186);