Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519075 |
| 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 | |
2088 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)(4,8)(5,6)(9,10)(11,17)(12,24)(13,27)(14,19)(15,32)(16,22)(18,31)(20,25)(21,37)(23,34)(26,33)(28,39)(29,35)(30,40)(36,38);(2,6)(7,8)(9,11,14,22)(10,16,19,17)(12,25,23,15)(13,28,30,21)(18,33,36,35)(20,24,32,34)(26,31,29,38)(27,37,40,39);(1,3,5,4)(2,7,6,8)(9,12,15,11,14,23,25,22)(10,17,20,24,19,16,32,34)(13,29,31,28,30,26,38,21)(18,33,27,39,36,35,40,37);(1,4,5,3)(2,8,6,7)(9,13,15,31,14,30,25,38)(10,18,20,27,19,36,32,40)(11,21,23,29,22,28,12,26)(16,33,34,39,17,35,24,37);(2,6)(7,8)(10,19)(16,17)(18,36)(20,32)(24,34)(27,40)(33,35)(37,39)Polycyclic presentation:
2^9|b^2=i,c^2=fgi,d^2=fgi,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=di,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7469009);