Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4670 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,4)(3,7,5,8)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24);(1,3,6,5)(2,7,4,8)(10,14)(15,17)(16,21)(22,23)(25,26,30,31)(27,29,33,38)(28,35,37,32)(34,40,39,36);(1,3,6,5)(2,8,4,7)(12,20)(16,21)(18,24)(22,23)(26,31)(29,38)(32,35)(36,40);(1,4,6,2)(3,7,5,8)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23)(25,27,30,33)(26,29,31,38)(28,34,37,39)(32,36,35,40);(1,5,6,3)(2,7,4,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,28,30,37)(26,32,31,35)(27,34,33,39)(29,36,38,40);(9,12,13,20)(10,16,14,21)(11,18,19,24)(15,22,17,23)(25,29)(26,33)(27,31)(28,36)(30,38)(32,39)(34,35)(37,40)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=h,d^2=hi,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8135825);