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 | |
1750 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)(9,10,16,21)(11,22,26,35)(12,27,29,34)(13,19,32,40)(14,20,15,33)(17,25,36,24)(18,23,37,28)(30,38,31,39);(1,2,5,6)(3,4,8,7)(9,11)(10,17)(12,13)(14,24)(15,25)(16,26)(18,19)(20,35)(21,36)(22,33)(23,30)(27,38)(28,31)(29,32)(34,39)(37,40);(1,3)(2,7)(4,6)(5,8)(9,12,14,23,16,29,15,28)(10,18,33,34,21,37,20,27)(11,13,24,30,26,32,25,31)(17,19,22,39,36,40,35,38);(2,6)(3,8)(9,13)(10,19)(11,23)(12,25)(14,30)(15,31)(16,32)(17,34)(18,35)(20,38)(21,40)(22,37)(24,29)(26,28)(27,36)(33,39);(1,4,5,7)(2,3,6,8)(9,14,16,15)(10,20,21,33)(11,24,26,25)(12,23,29,28)(13,30,32,31)(17,35,36,22)(18,27,37,34)(19,38,40,39)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=gi,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, 7092644);