Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470067 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #62852 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4232 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,5)(4,6)(7,8)(9,10,16,22)(11,23,26,40)(12,27,29,35)(13,19,32,39)(14,20,15,21)(17,24,34,25)(18,33,36,28)(30,37,31,38);(1,3)(2,5)(4,7)(6,8)(9,11)(10,17)(12,13)(14,24)(15,25)(16,26)(18,19)(20,23)(21,40)(22,34)(27,38)(28,30)(29,32)(31,33)(35,37)(36,39);(3,7)(5,8)(9,12)(10,18)(11,13)(14,28)(15,33)(16,29)(17,19)(20,35)(21,27)(22,36)(23,37)(24,30)(25,31)(26,32)(34,39)(38,40);(3,7)(5,8)(9,13)(10,19)(11,12)(14,31)(15,30)(16,32)(17,18)(20,38)(21,37)(22,39)(23,27)(24,33)(25,28)(26,29)(34,36)(35,40);(2,6)(5,8)(9,14,16,15)(10,20,22,21)(11,24,26,25)(12,28,29,33)(13,30,32,31)(17,23,34,40)(18,35,36,27)(19,37,39,38)Polycyclic presentation:
2^9|a^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,f^b=fi,g^c=gi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6377268);