Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419792 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | #41 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
917 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,5)(6,7)(9,10,15,20)(11,21,24,17)(12,16,23,34)(13,26,30,36)(14,31,32,19)(18,29,37,28)(22,39,40,33)(25,38,27,35);(1,2,4,5)(3,7,8,6)(9,11,15,24)(10,16,20,34)(12,14,23,32)(13,27,30,25)(17,31,21,19)(18,35,37,38)(22,29,40,28)(26,33,36,39);(9,12,14,24,15,23,32,11)(10,17,19,16,20,21,31,34)(13,25,29,40,30,27,28,22)(18,35,36,33,37,38,26,39);(1,3,4,8)(2,6,5,7)(9,13,15,30)(10,18,20,37)(11,22,24,40)(12,25,23,27)(14,29,32,28)(16,33,34,39)(17,35,21,38)(19,36,31,26)Polycyclic presentation:
2^9|a^2=i,b^2=ef,c^2=g,d^2=ef,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^c=hi,e^d=ei,f^d=fi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 351594);