Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24289 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2366 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1063 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,8)(4,6)(5,7)(9,10,14,19,15,20,13,17)(11,21,22,32,25,31,24,16)(12,26,28,34,29,33,27,18)(23,37,36,40,39,30,38,35);(1,3,5,4)(2,6,7,8)(9,11)(10,16,20,32)(12,23)(13,24)(14,22)(15,25)(17,31,19,21)(18,30,34,37)(26,35,33,40)(27,38)(28,36)(29,39);(1,4,5,3)(2,6,7,8)(10,17,20,19)(11,22,25,24)(18,33,34,26)(23,36,39,38);(2,7)(6,8)(9,12,13,27,15,29,14,28)(10,18,19,26,20,34,17,33)(11,23,24,38,25,39,22,36)(16,30,21,35,32,37,31,40)Polycyclic presentation:
2^9|a^2=h,b^2=eh,c^2=eg,d^2=f,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,g^b=gi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 276932);