Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519063 |
| 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 |
| prime radical | #61199 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1814 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,6)(3,7,4,8)(9,10,15,19)(11,16,22,32)(12,23,21,35)(13,26,29,40)(14,30,24,18)(17,33,34,28)(20,36,27,39)(25,38,37,31);(2,6)(7,8)(9,11,15,22)(10,16,19,32)(12,24,21,14)(13,27,29,20)(17,31,34,38)(18,35,30,23)(25,28,37,33)(26,39,40,36);(1,3,5,4)(2,7,6,8)(9,12,14,11,15,21,24,22)(10,16,18,23,19,32,30,35)(13,25,28,20,29,37,33,27)(17,31,26,36,34,38,40,39);(9,13,14,28,15,29,24,33)(10,17,18,26,19,34,30,40)(11,20,21,37,22,27,12,25)(16,31,23,36,32,38,35,39);(1,4,5,3)(2,7,6,8)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=fg,d^2=g,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^d=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7471633);