Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573643 |
| 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 | |
1918 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,5)(3,6,7,8)(9,10,15,22)(11,24,27,18)(12,28,26,37)(13,20,33,38)(14,34,25,21)(16,35,17,23)(19,31,39,30)(29,40,32,36);(2,5)(6,8)(9,11)(10,18)(12,29)(13,16)(14,30)(15,27)(17,33)(19,21)(20,23)(22,24)(25,31)(26,32)(28,36)(34,39)(35,38)(37,40);(9,12,15,26)(10,19,22,39)(11,25,27,14)(13,32,33,29)(16,31,17,30)(18,36,24,40)(20,21,38,34)(23,37,35,28);(1,3,4,7)(2,6,5,8)(9,13,15,33)(10,20,22,38)(11,16,27,17)(12,14,26,25)(18,23,24,35)(19,36,39,40)(21,28,34,37)(29,30,32,31);(9,14,15,25)(10,21,22,34)(11,26,27,12)(13,31,33,30)(16,32,17,29)(18,37,24,28)(19,38,39,20)(23,36,35,40)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=f,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,e^b=eh,h^b=hi,d^c=dh,g^c=gi,h^c=hi,e^d=eh,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6483297);