Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7563346 |
| 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 | #61407 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1742 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)(3,7)(4,5)(6,8)(9,10)(11,15)(12,23)(13,17)(14,19)(16,26)(18,28)(20,33)(21,30)(22,32)(24,29)(25,35)(27,31)(34,36)(37,40)(38,39);(2,5)(6,7)(9,11,14,22)(10,15,19,32)(12,24,25,38)(13,27,28,21)(16,33,34,40)(17,31,18,30)(20,36,37,26)(23,29,35,39);(9,12,13,26,14,25,28,36)(10,16,18,23,19,34,17,35)(11,20,21,24,22,37,27,38)(15,29,31,33,32,39,30,40);(10,17,19,18)(12,25)(15,30,32,31)(16,23,34,35)(20,37)(24,38)(26,36)(29,33,39,40);(1,3,4,8)(2,6,5,7)Polycyclic presentation:
2^9|b^2=i,c^2=g,d^2=h,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,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7521055);