Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418500 |
| 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 | |
1521 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)(9,10,16,22)(11,23,28,36)(12,29,32,18)(13,33,24,38)(14,21,15,20)(17,27,37,26)(19,34,40,30)(25,39,31,35);(1,2)(3,4)(9,11,15,27,16,28,14,26)(10,17,21,36,22,37,20,23)(12,30,25,24,32,34,31,13)(18,38,39,19,29,33,35,40);(2,4)(9,12,14,25,16,32,15,31)(10,18,20,39,22,29,21,35)(11,24,26,34,28,13,27,30)(17,19,23,33,37,40,36,38);(6,8)(9,13)(10,19)(11,25)(12,27)(14,30)(15,34)(16,24)(17,35)(18,23)(20,33)(21,38)(22,40)(26,32)(28,31)(29,36)(37,39)Polycyclic presentation:
2^9|a^2=i,b^2=ei,c^2=ef,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^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 330446);