Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993875 |
| 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 | |
1766 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,5)(4,6)(7,8)(9,10,14,20,16,11,15,19)(12,23,26,21,27,34,25,17)(13,18,30,28,31,22,32,36)(24,37,39,35,29,40,38,33);(1,2)(3,7)(4,6)(5,8)(9,11,16,10)(12,17,27,21)(13,28,31,36)(14,19,15,20)(18,30,22,32)(23,25,34,26)(24,37,29,40)(33,39,35,38);(1,3,4,8)(2,5,6,7)(9,12,16,27)(10,17,11,21)(13,29,31,24)(14,25,15,26)(18,35,22,33)(19,23,20,34)(28,37,36,40)(30,39,32,38);(3,8)(5,7)(9,13,16,31)(10,18,11,22)(12,24,27,29)(14,30,15,32)(17,33,21,35)(19,36,20,28)(23,37,34,40)(25,38,26,39);(2,6)(5,7)(9,14,16,15)(10,19,11,20)(12,25,27,26)(13,30,31,32)(17,23,21,34)(18,36,22,28)(24,38,29,39)(33,37,35,40)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=h,d^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,g^c=gi,f^d=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6970294);