Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417778 |
| 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 | C22×#135 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1576 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(5,6)(7,8)(9,10,15,20)(11,21,25,31)(12,26,28,36)(13,19,14,18)(16,22,33,24)(17,29,35,30)(23,32,38,39)(27,40,37,34);(1,2)(3,4)(9,11)(10,16)(12,27)(13,22)(14,24)(15,25)(17,34)(18,31)(19,21)(20,33)(23,29)(26,39)(28,37)(30,38)(32,36)(35,40);(2,4)(11,22,25,24)(12,28)(16,31,33,21)(17,35)(23,27,38,37)(26,36)(29,30)(32,34,39,40);(6,8)(9,12,13,29,15,28,14,30)(10,17,18,36,20,35,19,26)(11,23,24,27,25,38,22,37)(16,32,21,34,33,39,31,40)Polycyclic presentation:
2^9|a^2=i,c^2=f,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=di62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 330831);