Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682628 |
| 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 | |
2168 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)(11,21)(12,25)(13,18)(14,17)(15,20)(16,23)(19,26)(22,38)(24,36)(27,34)(28,40)(29,33)(30,37)(31,35)(32,39);(1,3)(2,5)(4,8)(6,7)(9,11,14,23)(10,16,17,21)(12,26,24,15)(13,28,29,27)(18,34,33,40)(19,25,20,36)(22,31,39,30)(32,35,38,37);(9,12,26,11,14,24,15,23)(10,16,20,36,17,21,19,25)(13,27,31,22,29,28,30,39)(18,32,37,40,33,38,35,34);(2,6)(5,7)(10,17)(16,21)(18,33)(19,20)(25,36)(32,38)(34,40)(35,37);(3,8)(5,7)(9,13)(10,18)(11,22)(12,27)(14,29)(15,30)(16,32)(17,33)(19,35)(20,37)(21,38)(23,39)(24,28)(25,34)(26,31)(36,40)Polycyclic presentation:
2^9|b^2=i,c^2=gi,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,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6589828);