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 |
| prime radical | C22×D4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1593 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)(11,19)(12,23)(13,27)(14,18)(15,20)(16,26)(17,28)(21,32)(22,34)(24,29)(25,39)(30,36)(31,37)(33,35)(38,40);(1,3,5,4)(2,6,8,7)(11,20)(13,28)(15,19)(16,31)(17,27)(21,35)(22,36)(24,40)(26,37)(29,38)(30,34)(32,33);(9,11,13,20,14,22,28,36)(10,15,17,19,18,30,27,34)(12,24,26,40,25,21,37,35)(16,32,23,38,31,29,39,33);(9,12)(10,16,18,31)(11,21)(13,26)(14,25)(15,29,30,32)(17,23,27,39)(19,33,34,38)(20,35)(22,24)(28,37)(36,40);(1,4,5,3)(2,7,8,6)(12,25)(16,31)(21,24)(23,39)(26,37)(29,32)(33,38)(35,40)Polycyclic presentation:
2^9|a^2=f,b^2=f,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=di,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7521321);