Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806827 |
| 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 | |
2017 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)(3,7)(4,6)(5,8)(9,10)(11,17)(12,25)(13,28)(14,18)(15,21)(16,23)(19,30)(20,26)(22,32)(24,33)(27,40)(29,35)(31,36)(34,39)(37,38);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,26)(13,29)(14,23)(15,24)(17,18)(19,35)(20,33)(21,25)(22,38)(27,30)(28,40)(31,39)(32,36)(34,37);(3,8)(5,7)(9,12,26,11,14,24,15,23)(10,17,20,33,18,16,21,25)(13,27,38,22,30,29,31,39)(19,34,36,40,28,32,37,35);(2,6)(5,7)(10,18)(16,17)(19,28)(20,21)(25,33)(32,34)(35,40)(36,37);(9,13,15,31,14,30,26,38)(10,19,21,37,18,28,20,36)(11,22,12,27,23,39,24,29)(16,32,33,40,17,34,25,35)Polycyclic presentation:
2^9|c^2=fh,e^2=fhi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,e^b=efh,h^b=hi,g^c=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6717240);