Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996405 |
| 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 | |
1774 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,6)(4,7)(9,10)(11,23)(12,29)(13,19)(14,20)(15,21)(16,22)(17,27)(18,30)(24,39)(25,37)(26,34)(28,35)(31,36)(32,38)(33,40);(1,2,5,6)(3,4,8,7)(9,11)(10,17)(12,24)(13,31)(14,26)(15,27)(16,28)(18,19)(20,23)(21,35)(22,34)(25,33)(29,38)(30,32)(36,40)(37,39);(1,3)(2,7)(4,6)(5,8)(9,12,15,30,16,31,14,25)(10,18,20,29,22,37,21,36)(11,24,27,32,28,13,26,33)(17,19,23,38,34,39,35,40);(2,6)(3,8)(9,13)(10,19)(11,25)(12,27)(14,32)(15,33)(16,24)(17,29)(18,35)(20,38)(21,40)(22,39)(23,37)(26,31)(28,30)(34,36);(1,4,5,7)(2,3,6,8)(9,14,16,15)(10,20,22,21)(11,26,28,27)(12,25,31,30)(13,32,24,33)(17,23,34,35)(18,29,37,36)(19,38,39,40)Polycyclic presentation:
2^9|b^2=h,c^2=g,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi,f^d=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7092641);