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)(9,10)(11,23)(12,16)(13,30)(14,19)(15,21)(17,22)(18,27)(20,33)(24,28)(25,34)(26,38)(29,35)(31,40)(32,37)(36,39);(1,3)(2,5)(4,7)(6,8)(9,11,15,27,17,29,16,28)(10,18,12,23,22,24,21,35)(13,31,14,26,33,39,32,25)(19,36,30,38,37,40,20,34);(9,12,16,10,17,21,15,22)(11,24,27,35,29,18,28,23)(13,20,32,37,33,30,14,19)(25,34,31,36,26,38,39,40);(3,7)(5,8)(9,13,15,14,17,33,16,32)(10,19,12,30,22,37,21,20)(11,25,28,39,29,26,27,31)(18,34,35,40,24,38,23,36);(9,14,16,13,17,32,15,33)(10,20,21,37,22,30,12,19)(11,26,28,31,29,25,27,39)(18,34,35,40,24,38,23,36)Polycyclic presentation:
2^9|a^2=gh,b^2=f,c^2=gi,d^2=g,e^2=fi,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, 7500215);