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 | |
4603 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,19)(12,17)(13,18)(14,15)(16,20)(21,22)(23,24)(25,26)(27,31)(28,36)(29,30)(32,37)(33,34)(35,39)(38,40);(2,5)(7,8)(10,15)(13,23)(16,19)(17,21)(25,26)(27,33)(28,36)(29,30)(31,34)(32,37)(35,38)(39,40);(1,2,6,5)(3,8,4,7)(11,20)(12,22)(16,19)(17,21)(26,30)(27,34)(32,36)(35,40);(1,3,6,4)(2,7,5,8)(9,11)(10,16)(12,18)(13,21)(14,20)(15,19)(17,23)(22,24)(25,26)(27,31)(28,32)(29,30)(33,34)(35,38)(36,37)(39,40);(1,4,6,3)(2,8,5,7)(9,12)(10,17)(11,18)(13,19)(14,22)(15,21)(16,23)(20,24)(25,27,29,34)(26,31,30,33)(28,35,37,40)(32,38,36,39);(1,5,6,2)(3,8,4,7)(9,13,14,23)(10,18,15,24)(11,21,20,17)(12,19,22,16)(25,28,29,37)(26,32,30,36)(27,35,34,40)(31,38,33,39)Polycyclic presentation:
2^9|c^2=hi,d^2=hi,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, 8731363);