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 | |
4634 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)(3,8)(4,7)(5,6)(9,10)(11,17)(12,20)(13,14)(15,18)(16,21)(19,22)(23,24)(26,28)(31,35)(32,36)(39,40);(3,4)(7,8)(10,14)(15,17)(16,20)(22,23)(25,26,30,28)(27,31,34,35)(29,32,38,36)(33,39,37,40);(1,3)(2,7)(4,6)(5,8)(10,14)(11,18)(12,21)(22,23)(25,27,30,34)(26,31,28,35)(29,37,38,33)(32,40,36,39);(3,4)(7,8)(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,23)(21,24)(25,28,30,26)(27,31,34,35)(29,36,38,32)(33,39,37,40);(1,4,6,3)(2,7,5,8)(9,10)(11,17)(12,16)(13,14)(15,18)(19,23)(20,21)(22,24)(25,27)(26,31)(28,35)(29,33)(30,34)(32,39)(36,40)(37,38);(1,5,6,2)(3,7,4,8)(9,12,13,21)(10,16,14,20)(11,19,18,24)(15,22,17,23)(25,29)(26,32)(27,33)(28,36)(30,38)(31,39)(34,37)(35,40)Polycyclic presentation:
2^9|b^2=ghi,c^2=ghi,d^2=ghi,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8923834);