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 | |
4630 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,7)(5,8)(13,21)(17,22)(18,23)(19,24)(25,26)(27,33)(28,32)(29,37)(30,40)(31,34)(35,38)(36,39);(1,2,4,6)(3,5,7,8)(9,10)(11,12)(13,17)(14,15)(16,20)(18,19)(21,22)(23,24)(25,27,31,37)(26,29,34,33)(28,35,30,36)(32,39,40,38);(1,3)(2,5)(4,7)(6,8)(10,15)(11,16)(17,22)(18,23)(25,28,31,30)(26,32,34,40)(27,35,37,36)(29,39,33,38);(1,2)(3,5)(4,6)(7,8)(9,11,14,16)(10,12,15,20)(13,18,21,23)(17,19,22,24)(25,29,31,33)(26,27,34,37)(28,38,30,39)(32,36,40,35);(1,3)(2,5)(4,7)(6,8)(9,12)(10,16)(11,15)(13,19)(14,20)(17,23)(18,22)(21,24)(25,27)(26,33)(28,35)(29,34)(30,36)(31,37)(32,38)(39,40);(9,13)(10,17)(11,18)(12,19)(14,21)(15,22)(16,23)(20,24)(25,30,31,28)(26,32,34,40)(27,36,37,35)(29,39,33,38)Polycyclic presentation:
2^9|b^2=gh,c^2=g,d^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8019723);