Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470058 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4223 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)(11,21)(12,26)(13,18)(14,19)(15,20)(16,23)(17,27)(22,30)(24,32)(25,31)(28,33)(29,34)(35,37)(36,39)(38,40);(1,3)(2,5)(4,7)(6,8)(9,11)(10,16)(12,22)(13,24)(14,23)(15,25)(17,30)(18,21)(19,31)(20,32)(26,39)(27,37)(28,38)(29,36)(33,35)(34,40);(3,7)(5,8)(9,12)(10,17)(11,22)(13,27)(14,29)(15,28)(16,30)(18,33)(19,26)(20,34)(21,35)(23,36)(24,37)(25,38)(31,39)(32,40);(3,7)(5,8)(9,13,15,14)(10,18,20,19)(11,23,25,24)(12,27,28,29)(16,31,32,21)(17,33,34,26)(22,36,38,37)(30,39,40,35);(2,6)(5,8)(9,13,15,14)(10,18,20,19)(11,24,25,23)(12,27,28,29)(16,21,32,31)(17,33,34,26)(22,37,38,36)(30,35,40,39)Polycyclic presentation:
2^9|d^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=dhi,f^b=fi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6377262);