Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514172 |
| 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 | C2×2−1+4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1648 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,8)(3,7,4,6)(9,10,15,19)(11,20,24,16)(12,25,17,30)(13,27,31,28)(14,32,34,18)(21,22,40,29)(23,37,33,38)(26,39,36,35);(1,3,5,4)(2,6,8,7)(9,11,15,24)(10,16,19,20)(12,26,17,36)(13,29,31,22)(14,33,34,23)(18,38,32,37)(21,27,40,28)(25,35,30,39);(9,12)(10,13)(11,21)(14,27)(15,17)(16,35)(18,30)(19,31)(20,39)(22,37)(23,36)(24,40)(25,32)(26,33)(28,34)(29,38);(9,13,15,31)(10,17,19,12)(11,22,24,29)(14,30,34,25)(16,36,20,26)(18,28,32,27)(21,37,40,38)(23,39,33,35);(1,4,5,3)(2,7,8,6)(9,14,15,34)(10,18,19,32)(11,23,24,33)(12,27,17,28)(13,30,31,25)(16,37,20,38)(21,36,40,26)(22,39,29,35)Polycyclic presentation:
2^9|a^2=f,b^2=f,d^2=i,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,e^b=ei,g^b=gi,g^c=gi,h^c=hi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7426511);