Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7563327 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1656 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,6)(3,5,7,8)(10,17)(11,23)(12,13)(14,15)(18,34)(19,20)(21,22)(25,30)(27,28)(31,32)(35,40)(37,39);(1,3)(2,5)(4,7)(6,8)(9,10,14,17,16,22,15,21)(11,24,27,34,28,36,23,18)(12,29,31,19,32,33,13,20)(25,35,26,39,30,37,38,40);(9,11,14,27,16,28,15,23)(10,18,21,36,22,34,17,24)(12,30,31,38,32,25,13,26)(19,37,29,39,20,35,33,40);(3,7)(5,8)(9,12,15,13,16,32,14,31)(10,19,17,33,22,20,21,29)(11,25,23,38,28,30,27,26)(18,35,24,39,34,37,36,40);(9,13,14,12,16,31,15,32)(10,20,17,29,22,19,21,33)(11,26,27,30,28,38,23,25)(18,35,24,39,34,37,36,40)Polycyclic presentation:
2^9|a^2=gh,b^2=f,c^2=gi,d^2=g,e^2=f,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7500213);