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 | |
4648 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10,15,18)(11,16,12,13)(14,17,22,24)(19,23,20,21)(25,26)(27,32)(28,36)(29,37)(30,35)(31,33)(34,38)(39,40);(1,3,5,8)(2,6,4,7)(9,11,15,12)(10,13,18,16)(14,20,22,19)(17,23,24,21)(25,27,31,36)(26,32,33,28)(29,30,38,39)(34,40,37,35);(11,12)(13,16)(19,20)(21,23)(26,33)(27,36)(29,38)(35,40);(1,2,5,4)(3,6,8,7)(9,12,15,11)(10,16,18,13)(14,20,22,19)(17,23,24,21)(25,28)(26,27)(29,35)(30,34)(31,32)(33,36)(37,39)(38,40);(1,4,5,2)(3,7,8,6)(9,13,15,16)(10,11,18,12)(14,21,22,23)(17,19,24,20)(25,29)(26,34)(27,30)(28,35)(31,38)(32,40)(33,37)(36,39);(9,14)(10,17)(11,19)(12,20)(13,21)(15,22)(16,23)(18,24)(25,30)(26,35)(27,29)(28,34)(31,39)(32,37)(33,40)(36,38)Polycyclic presentation:
2^9|a^2=g,b^2=gh,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7646715);