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 | |
4450 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(10,15)(11,16)(17,21)(18,23)(25,26)(27,31)(28,34)(29,37)(30,33)(32,38)(35,40)(36,39);(2,5)(7,8)(9,10,14,15)(11,12,16,19)(13,21,22,17)(18,24,23,20)(25,27,30,28)(26,31,33,34)(29,35,38,36)(32,39,37,40);(1,2,6,5)(3,8,4,7)(11,16)(12,19)(18,23)(20,24)(25,28,30,27)(26,31,33,34)(29,36,38,35)(32,39,37,40);(1,3,6,4)(2,7,5,8)(9,11,14,16)(10,12,15,19)(13,18,22,23)(17,20,21,24);(1,4,6,3)(2,8,5,7)(9,12)(10,16)(11,15)(13,20)(14,19)(17,23)(18,21)(22,24)(25,26)(27,34)(28,31)(29,32)(30,33)(35,40)(36,39)(37,38);(1,5,6,2)(3,8,4,7)(9,13,14,22)(10,17,15,21)(11,18,16,23)(12,20,19,24)(25,29)(26,32)(27,35)(28,36)(30,38)(31,39)(33,37)(34,40)Polycyclic presentation:
2^9|b^2=gh,c^2=hi,d^2=gi,e^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8731349);