Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667964 |
| 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 |
| prime radical | #23878 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1938 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,23)(13,21)(14,19)(16,18)(17,28)(20,25)(22,32)(24,27)(26,36)(29,30)(31,34)(33,39)(35,38)(37,40);(2,5)(3,7)(9,11,15,10)(12,24,27,23)(13,21,16,18)(14,30,32,34)(17,25,20,28)(19,31,22,29)(26,33,40,35)(36,38,37,39);(9,12,15,27)(10,17,11,20)(13,25,16,28)(14,26,32,40)(18,24,21,23)(19,33,22,35)(29,38,31,39)(30,36,34,37);(2,5)(6,8)(9,13,15,16)(10,18,11,21)(12,25,27,28)(14,31,32,29)(17,24,20,23)(19,30,22,34)(26,39,40,38)(33,36,35,37);(1,3,4,7)(2,6,5,8)(9,14)(10,19)(11,22)(12,26)(13,29)(15,32)(16,31)(17,33)(18,34)(20,35)(21,30)(23,36)(24,37)(25,38)(27,40)(28,39)Polycyclic presentation:
2^9|b^2=i,c^2=i,d^2=i,e^2=fi,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,g^c=gi,h^c=hi,e^d=ei,h^e=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6580359);