Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7563333 |
| 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 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1745 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)(11,21)(12,25)(13,16)(14,31)(15,20)(17,28)(18,23)(19,27)(22,36)(24,33)(26,37)(29,40)(30,34)(32,39)(35,38);(2,6)(7,8)(9,11,15,13)(10,16,20,21)(12,26,29,39)(14,23,27,24)(17,34,35,36)(18,31,33,19)(22,38,30,28)(25,32,40,37);(1,3,5,4)(2,7,6,8)(9,12,27,38,15,29,14,28)(10,17,31,40,20,35,19,25)(11,22,23,39,13,30,24,26)(16,32,18,36,21,37,33,34);(9,13,15,11)(10,18)(12,22,29,30)(14,23,27,24)(16,19)(17,34)(20,33)(21,31)(25,37)(26,38,39,28)(32,40)(35,36);(1,4,5,3)(2,7,6,8)(9,14,15,27)(10,19,20,31)(11,23,13,24)(12,28,29,38)(16,18,21,33)(17,25,35,40)(22,39,30,26)(32,36,37,34)Polycyclic presentation:
2^9|b^2=i,c^2=fg,d^2=hi,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=di,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7521468);