Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993899 |
| 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 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1898 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(3,7)(6,8)(9,10,15,19)(11,20,24,30)(12,25,29,34)(13,17,31,23)(14,22,32,18)(16,27,35,28)(21,37,39,26)(33,40,38,36);(2,5)(3,7)(9,11)(10,13)(12,21)(14,17)(15,24)(16,26)(18,30)(19,31)(20,22)(23,32)(25,36)(27,33)(28,38)(29,39)(34,40)(35,37);(1,2)(3,6)(4,5)(7,8)(9,12)(10,16)(11,21)(13,26)(14,27)(15,29)(17,33)(18,34)(19,35)(20,36)(22,25)(23,38)(24,39)(28,32)(30,40)(31,37);(2,5)(6,8)(9,13)(10,17)(11,22)(12,26)(14,30)(15,31)(16,33)(18,24)(19,23)(20,32)(21,25)(27,40)(28,36)(29,37)(34,39)(35,38);(1,3,4,7)(2,6,5,8)(9,14,15,32)(10,18,19,22)(11,17,24,23)(12,27,29,28)(13,30,31,20)(16,34,35,25)(21,33,39,38)(26,40,37,36)Polycyclic presentation:
2^9|a^2=i,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,g^c=gi,f^d=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7050416);