Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #114839 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1438 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1105 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)(3,4,8,6)(5,7)(9,10,14,19,15,20,13,18)(11,21,25,16,24,32,23,31)(12,17,28,35,29,36,27,34)(22,37,26,40,39,33,38,30);(1,3)(2,6)(4,7)(5,8)(9,11)(10,16)(12,26)(13,25)(14,23)(15,24)(17,33)(18,32)(19,21)(20,31)(22,28)(27,39)(29,38)(30,34)(35,40)(36,37);(1,4,5,6)(2,3,7,8)(9,12)(10,17)(11,22)(13,27)(14,28)(15,29)(16,30)(18,34)(19,35)(20,36)(21,33)(23,38)(24,39)(25,26)(31,40)(32,37)Polycyclic presentation:
2^9|a^2=g,c^2=h,d^2=i,f^2=i,g^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=ehi,f^b=fi,g^b=ghi,d^c=dh,f^c=fi,g^c=gh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 22040);