Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437602 |
| 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 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2008 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,7)(3,8,4,6)(9,10,14,19)(11,21,24,34)(12,17,28,32)(13,18,15,20)(16,23,31,25)(22,35,37,40)(26,29,33,27)(30,36,39,38);(1,3,5,4)(2,6,7,8)(9,10)(11,21)(12,26)(13,20)(14,19)(15,18)(16,23)(17,27)(22,30)(24,34)(25,31)(28,33)(29,32)(35,38)(36,40)(37,39);(1,4,5,3)(2,6,7,8)(9,11,14,24)(10,16,19,31)(12,22,28,37)(13,23,15,25)(17,30,32,39)(18,21,20,34)(26,35,33,40)(27,36,29,38);(9,12)(10,17)(11,22)(13,27)(14,28)(15,29)(16,30)(18,26)(19,32)(20,33)(21,35)(23,36)(24,37)(25,38)(31,39)(34,40);(9,13,14,15)(10,18,19,20)(11,23,24,25)(12,27,28,29)(16,21,31,34)(17,26,32,33)(22,36,37,38)(30,35,39,40)Polycyclic presentation:
2^9|a^2=f,b^2=fi,c^2=f,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^b=ei,h^b=hi,g^c=gi,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6346455);