Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417784 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1590 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,4)(9,10)(11,23)(12,27)(13,19)(14,21)(15,20)(16,22)(17,25)(18,31)(24,39)(26,34)(28,35)(29,33)(30,37)(32,36)(38,40);(5,6)(7,8)(9,11,14,25,16,13,15,26)(10,17,20,23,22,34,21,19)(12,28,30,29,32,40,31,24)(18,35,36,39,37,38,27,33);(1,2,3,4)(6,8)(9,12)(10,18)(11,24)(13,29)(14,31)(15,30)(16,32)(17,33)(19,35)(20,27)(21,36)(22,37)(23,38)(25,40)(26,28)(34,39);(9,13,16,11)(10,19,22,23)(12,29,32,24)(14,25,15,26)(17,20,34,21)(18,35,37,38)(27,39,36,33)(28,31,40,30)Polycyclic presentation:
2^9|b^2=e,c^2=ef,d^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=dei,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 330410);