Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682628 |
| 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 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2218 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(3,7)(9,10)(11,16)(12,25)(13,18)(14,20)(15,21)(17,24)(19,30)(22,31)(23,32)(26,34)(27,37)(28,35)(29,36)(33,38)(39,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,14)(13,26)(15,24)(17,21)(18,34)(19,32)(20,25)(22,28)(23,30)(27,38)(29,39)(31,35)(33,37)(36,40);(9,12,14,24,15,23,30,11)(10,17,19,25,21,16,20,32)(13,26,27,38,29,39,28,22)(18,33,35,34,36,31,37,40);(1,3,4,7)(2,6,5,8);(2,5)(6,8)(9,13)(10,18)(11,22)(12,26)(14,27)(15,29)(16,31)(17,33)(19,35)(20,37)(21,36)(23,39)(24,38)(25,34)(28,30)(32,40)Polycyclic presentation:
2^9|c^2=g,d^2=f,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, 6590936);