Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7563319 |
| 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 | |
1662 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10)(11,25)(12,32)(13,19)(14,20)(15,23)(16,22)(17,24)(18,29)(21,33)(26,34)(27,35)(28,38)(30,37)(31,36)(39,40);(1,3)(2,5)(4,7)(6,8)(9,11,15,29,17,31,16,30)(10,18,22,25,24,37,23,36)(12,27,13,26,14,28,33,40)(19,38,32,34,21,35,20,39);(9,12,15,13,17,14,16,33)(10,19,22,32,24,21,23,20)(11,26,30,27,31,40,29,28)(18,34,36,38,37,39,25,35);(3,7)(5,8)(9,13,16,12,17,33,15,14)(10,20,23,21,24,32,22,19)(11,27,29,26,31,28,30,40)(18,35,25,39,37,38,36,34);(9,14,15,33,17,12,16,13)(10,21,22,20,24,19,23,32)(11,28,29,40,31,27,30,26)(18,35,25,39,37,38,36,34)Polycyclic presentation:
2^9|a^2=gh,b^2=f,c^2=g,d^2=gi,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=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7500208);