Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #112870 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #390 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1445 |
| prime radical | C25 | #41 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
686 conjugacy classes of subgroups.
| 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)(11,19,22,32)(12,16)(13,17)(14,18)(15,21,28,25)(20,35,37,34)(23,30)(24,31)(26,33)(27,38,29,36)(39,40);(1,3)(2,5)(4,8)(6,7)(9,11,13,25,14,22,26,21)(10,15,17,32,18,28,33,19)(12,20,23,38,24,37,39,36)(16,29,30,34,31,27,40,35);(1,3)(2,5)(4,8)(6,7)(9,12,13,23,14,24,26,39)(10,16,17,30,18,31,33,40)(11,20,25,38,22,37,21,36)(15,27,32,35,28,29,19,34)Polycyclic presentation:
2^9|a^2=g,b^2=de,c^2=de,d^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,c^b=cf,d^b=di,e^b=ei,g^b=ghi,g^c=gh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 28956);