Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682648 |
| 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 | |
1831 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,7)(3,8,4,6)(9,10,15,21)(11,16,25,36)(12,26,28,39)(13,18,32,27)(14,20,35,19)(17,31,37,30)(22,33,40,34)(23,38,24,29);(1,2)(3,6)(4,8)(5,7)(9,11)(10,16)(12,27)(13,26)(14,23)(15,25)(17,34)(18,28)(19,29)(20,38)(21,36)(22,31)(24,35)(30,40)(32,39)(33,37);(9,12,14,30,15,28,35,31)(10,17,19,26,21,37,20,39)(11,22,23,27,25,40,24,18)(13,29,33,36,32,38,34,16);(1,3,5,4)(2,6,7,8)(12,28)(13,32)(17,37)(18,27)(22,40)(26,39)(30,31)(33,34);(1,4,5,3)(2,6,7,8)(9,13)(10,18)(11,17)(12,29)(14,33)(15,32)(16,31)(19,22)(20,40)(21,27)(23,26)(24,39)(25,37)(28,38)(30,36)(34,35)Polycyclic presentation:
2^9|a^2=fi,c^2=g,d^2=f,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,e^d=ei,h^e=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6594196);