Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454681 |
| 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 | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
751 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(3,7)(5,8)(9,10,14,16,15,20,13,19)(11,21,22,17,25,31,24,34)(12,26,28,18,30,32,29,36)(23,38,27,37,40,33,39,35);(1,2,4,6)(3,8,7,5)(10,16)(11,22)(13,14)(18,32)(19,20)(21,31)(23,27)(24,25)(26,36)(28,29)(35,37)(39,40);(1,3,4,7)(2,5,6,8)(9,11,14,22,15,25,13,24)(10,17,19,21,20,34,16,31)(12,27,29,23,30,39,28,40)(18,35,32,38,36,37,26,33);(9,12)(10,18)(11,23)(13,29)(14,28)(15,30)(16,32)(17,33)(19,26)(20,36)(21,37)(22,27)(24,39)(25,40)(31,35)(34,38)Polycyclic presentation:
2^9|a^2=efg,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,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=de,e^d=ei,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, 381941);