Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417603 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×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 | |
1892 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)(6,8)(9,10)(11,23)(12,18)(13,32)(14,21)(15,20)(16,22)(17,26)(19,30)(24,35)(25,38)(27,37)(28,36)(29,34)(31,39)(33,40);(1,2,3,4)(9,11,14,26,16,28,15,27)(10,17,20,23,22,37,21,36)(12,29,30,25,13,33,31,24)(18,38,39,34,32,35,19,40);(5,6,7,8)(9,12,16,13)(10,18,22,32)(11,24,28,25)(14,30,15,31)(17,34,37,40)(19,20,39,21)(23,35,36,38)(26,29,27,33);(2,4)(9,13,16,12)(10,19,22,39)(11,25,28,24)(14,30,15,31)(17,35,37,38)(18,20,32,21)(23,34,36,40)(26,29,27,33)Polycyclic presentation:
2^9|b^2=e,c^2=fi,d^2=i,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,g^b=gi,g^c=gi,e^d=ei,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 330110);