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 | |
4613 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)(11,16)(12,17)(13,21)(14,15)(18,22)(19,20)(23,24)(25,26)(27,28)(29,37)(30,31)(32,38)(33,36)(34,40)(35,39);(1,3,5,8)(2,6,4,7)(10,15)(12,16)(18,21)(20,23)(25,26,30,31)(27,28,33,36)(29,32,38,37)(34,35,39,40);(1,2,5,4)(3,6,8,7)(10,15)(11,17)(18,21)(19,24)(26,31)(27,33)(29,38)(35,40);(9,11,14,17)(10,16,15,12)(13,19,22,24)(18,23,21,20)(25,27,30,33)(26,28,31,36)(29,34,38,39)(32,35,37,40);(9,12,14,16)(10,17,15,11)(13,20,22,23)(18,24,21,19)(25,28)(26,27)(29,35)(30,36)(31,33)(32,34)(37,39)(38,40);(1,4,5,2)(3,7,8,6)(9,13)(10,18)(11,19)(12,20)(14,22)(15,21)(16,23)(17,24)(25,29)(26,32)(27,34)(28,35)(30,38)(31,37)(33,39)(36,40)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,c^2=gh,d^2=h,e^2=hi,f^2=gh,b^a=bg,c^a=ch,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^c=fi,e^d=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7919251);