Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682658 |
| 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 | |
1787 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,17)(12,13)(14,18)(15,19)(16,20)(21,25)(22,31)(23,35)(24,32)(26,27)(28,33)(29,36)(30,34)(37,39)(38,40);(2,5)(3,7)(9,11)(10,17)(12,25)(13,21)(14,28)(15,23)(16,24)(18,33)(19,35)(20,32)(22,27)(26,31)(29,38)(30,37)(34,39)(36,40);(9,12,13,10,16,19,15,20)(11,17,21,35,24,32,23,25)(14,26,27,18,30,36,29,34)(22,31,37,33,38,40,28,39);(2,5)(6,8)(9,13,16,15)(10,12,20,19)(11,21,24,23)(14,27,30,29)(17,25,32,35)(18,26,34,36)(22,37,38,28)(31,39,40,33);(1,3,4,7)(2,6,5,8)(9,14)(10,18)(11,22)(12,26)(13,27)(15,29)(16,30)(17,31)(19,36)(20,34)(21,37)(23,28)(24,38)(25,39)(32,40)(33,35)Polycyclic presentation:
2^9|c^2=gi,d^2=i,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,h^e=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6592716);