Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470231 |
| 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 | C24 | C25 |
| prime radical | #62859 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×Q8 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
4142 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)(3,5)(4,6)(7,8)(9,10,15,14)(11,16,22,21)(12,17,25,24)(13,26,27,18)(19,31,33,28)(20,34,35,29)(23,36,37,30)(32,38,40,39);(1,3)(2,5)(4,7)(6,8)(9,11)(10,16)(12,19)(13,20)(14,21)(15,22)(17,28)(18,29)(23,32)(24,31)(25,33)(26,34)(27,35)(30,38)(36,39)(37,40);(3,7)(5,8)(9,12)(10,17)(11,19)(13,23)(14,24)(15,25)(16,28)(18,30)(20,32)(21,31)(22,33)(26,36)(27,37)(29,38)(34,39)(35,40);(3,7)(5,8)(9,13,15,27)(10,18,14,26)(11,20,22,35)(12,23,25,37)(16,29,21,34)(17,30,24,36)(19,32,33,40)(28,38,31,39);(2,6)(5,8)(9,14,15,10)(11,21,22,16)(12,24,25,17)(13,18,27,26)(19,31,33,28)(20,29,35,34)(23,30,37,36)(32,38,40,39)Polycyclic presentation:
2^9|a^2=i,d^2=i,e^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dh,g^b=gi,f^c=fi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6377444);