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 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1918 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)(3,7)(4,6)(5,8)(9,10)(11,20)(12,26)(13,30)(14,19)(15,18)(16,24)(17,28)(21,38)(22,23)(25,33)(27,32)(29,36)(31,35)(34,39)(37,40);(1,3,4,8)(2,5,6,7)(9,11)(10,16)(12,27)(13,25)(14,24)(15,22)(17,34)(18,33)(19,20)(21,28)(23,30)(26,37)(29,40)(31,39)(32,36)(35,38);(9,12,13,28,14,29,15,31)(10,17,18,26,19,35,30,36)(11,21,25,40,24,39,22,27)(16,32,33,34,20,37,23,38);(2,6)(5,7)(9,13,14,15)(10,18,19,30)(11,22,24,25)(12,28,29,31)(16,23,20,33)(17,26,35,36)(21,27,39,40)(32,38,37,34);(3,8)(5,7)(9,10)(11,23)(12,17)(13,18)(14,19)(15,30)(16,25)(20,22)(21,38)(24,33)(26,28)(27,37)(29,35)(31,36)(32,40)(34,39)Polycyclic presentation:
2^9|b^2=fi,c^2=g,d^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,h^b=hi,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, 6590465);