Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667964 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
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 | |
2606 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)(11,17)(12,26)(13,19)(14,20)(15,21)(16,22)(18,29)(23,39)(24,34)(25,36)(27,33)(28,37)(30,40)(31,38)(32,35);(1,3,4,8)(2,5,6,7)(9,11,16,25)(10,17,22,36)(12,27,30,31)(13,24,15,32)(14,28,23,29)(18,20,37,39)(19,34,21,35)(26,33,40,38);(9,12)(10,18)(11,23)(13,28)(14,25)(15,29)(16,30)(17,33)(19,26)(20,35)(21,40)(22,37)(24,31)(27,32)(34,39)(36,38);(2,6)(5,7)(9,13,16,15)(10,19,22,21)(11,24,25,32)(12,28,30,29)(14,27,23,31)(17,34,36,35)(18,26,37,40)(20,33,39,38);(3,8)(5,7)(9,14,16,23)(10,20,22,39)(11,12,25,30)(13,31,15,27)(17,26,36,40)(18,35,37,34)(19,38,21,33)(24,29,32,28)Polycyclic presentation:
2^9|b^2=fi,d^2=i,e^2=i,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, 6575819);