Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454664 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2622 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
832 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(9,10,13,18,14,19,30,17)(11,20,23,33,24,32,22,15)(12,25,27,36,29,35,28,16)(21,31,26,34,39,40,38,37);(1,2,4,6)(3,7,8,5)(9,11,14,24)(10,15,19,33)(12,26,29,38)(13,23,30,22)(16,34,36,37)(17,32,18,20)(21,27,39,28)(25,40,35,31);(1,3,4,8)(2,5,6,7)(9,12,13,28,14,29,30,27)(10,16,18,35,19,36,17,25)(11,21,22,26,24,39,23,38)(15,31,32,34,33,40,20,37);(10,17)(12,27)(13,30)(15,32)(18,19)(20,33)(21,38)(22,23)(25,35)(26,39)(28,29)(31,40)Polycyclic presentation:
2^9|a^2=f,b^2=gh,c^2=g,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,e^b=ei,g^b=gi,h^b=hi,d^c=de,f^c=fi,e^d=ei,f^d=fi,g^d=gi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 381785);