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 | |
2079 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(3,7)(6,8)(9,10,13,20,15,21,14,19)(11,22,24,17,25,34,32,16)(12,26,29,37,31,18,30,38)(23,39,27,33,28,36,40,35);(2,5)(3,7)(9,11,15,25)(10,16,21,17)(12,27,31,40)(13,32,14,24)(18,36,26,39)(19,22,20,34)(23,30,28,29)(33,37,35,38);(1,2)(3,6)(4,5)(7,8)(9,11)(10,17)(12,28)(13,24)(14,32)(15,25)(16,21)(18,33)(19,22)(20,34)(23,31)(26,35)(27,30)(29,40)(36,38)(37,39);(2,5)(6,8)(9,12,14,30,15,31,13,29)(10,18,19,37,21,26,20,38)(11,23,32,40,25,28,24,27)(16,33,34,39,17,35,22,36);(1,3,4,7)(2,6,5,8)(9,13,15,14)(10,19,21,20)(11,24,25,32)(12,29,31,30)(16,34,17,22)(18,37,26,38)(23,27,28,40)(33,39,35,36)Polycyclic presentation:
2^9|a^2=f,b^2=i,d^2=fi,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,g^c=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7053029);