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 | |
4934 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,7,8,4)(9,10,14,15)(11,19,20,16)(12,21,22,17)(13,23,24,18)(25,26,29,32)(27,33,35,30)(28,31,37,40)(34,39,36,38);(2,6)(4,7)(10,15)(13,24)(16,19)(17,21)(25,27,29,35)(26,30,32,33)(28,36,37,34)(31,39,40,38);(1,2)(3,4)(5,6)(7,8)(10,15)(11,20)(17,21)(18,23)(25,26,29,32)(27,30,35,33)(28,31,37,40)(34,38,36,39);(1,3)(2,4)(5,8)(6,7)(9,11)(10,16)(12,18)(13,17)(14,20)(15,19)(21,24)(22,23)(25,26)(27,30)(28,31)(29,32)(33,35)(34,38)(36,39)(37,40);(1,4)(2,3)(5,7)(6,8)(9,12)(10,17)(11,18)(13,16)(14,22)(15,21)(19,24)(20,23)(25,27)(26,30)(28,34)(29,35)(31,38)(32,33)(36,37)(39,40);(1,3)(2,4)(5,8)(6,7)(9,13)(10,18)(11,17)(12,16)(14,24)(15,23)(19,22)(20,21)(25,28)(26,31)(27,34)(29,37)(30,38)(32,40)(33,39)(35,36)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8659127);