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 | |
4631 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,7)(3,5,8,6)(25,26)(27,35)(28,37)(29,33)(30,39)(31,32)(34,40)(36,38);(1,3)(2,5)(4,8)(6,7)(9,10)(11,17)(12,20)(13,14)(15,18)(16,21)(19,23)(22,24)(25,27,31,37)(26,28,32,35)(29,30,36,40)(33,34,38,39);(1,3,4,8)(2,6,7,5)(10,14)(11,18)(12,21)(23,24)(26,32)(28,35)(29,36)(30,40);(3,8)(5,6)(9,11)(10,15)(12,22)(13,18)(14,17)(16,24)(19,21)(20,23)(25,28)(26,27)(29,39)(30,38)(31,35)(32,37)(33,40)(34,36);(1,3)(2,5)(4,8)(6,7)(9,12)(10,16)(11,19)(13,21)(14,20)(15,23)(17,24)(18,22)(25,29,31,36)(26,33,32,38)(27,30,37,40)(28,34,35,39);(1,2)(3,5)(4,7)(6,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,23)(20,21)(22,24)(25,30)(26,34)(27,36)(28,38)(29,37)(31,40)(32,39)(33,35)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8965415);