Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455962 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2826 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | 2−1+4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
614 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,5)(6,8)(9,10,14,21,16,20,15,29)(11,22,26,35,27,17,25,36)(12,13,30,33,19,32,31,18)(23,37,28,38,39,40,34,24);(1,2,4,5)(3,6,7,8)(9,11)(10,17)(12,28)(13,24)(14,25)(15,26)(16,27)(18,37)(19,34)(20,22)(21,35)(23,30)(29,36)(31,39)(32,38)(33,40);(1,3)(2,6)(4,7)(5,8)(9,12,16,19)(10,18,20,33)(11,23,27,39)(13,21,32,29)(14,31,15,30)(17,24,22,38)(25,28,26,34)(35,37,36,40);(3,7)(6,8)(9,13,14,33,16,32,15,18)(10,19,21,31,20,12,29,30)(11,24,25,40,27,38,26,37)(17,34,35,39,22,28,36,23)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=i,d^2=f,f^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,g^b=gi,h^b=hi,d^c=de,f^c=fi,g^c=gi,h^c=hi,e^d=ei,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 381429);