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 | |
4687 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,4)(3,8,5,7)(9,10)(11,17)(12,19)(13,21)(14,15)(16,23)(18,20)(22,24)(25,26)(27,32)(28,38)(29,39)(30,33)(31,37)(34,40)(35,36);(10,15)(12,17)(16,21)(20,24)(25,27)(26,31)(28,35)(29,36)(30,37)(32,33)(34,39)(38,40);(1,3,6,5)(2,7,4,8)(9,11)(10,12)(13,18)(14,19)(15,17)(16,20)(21,24)(22,23)(25,28)(26,29)(27,34)(30,39)(31,40)(32,36)(33,38)(35,37);(1,3,6,5)(2,8,4,7)(9,11)(10,12)(13,22)(14,19)(15,17)(16,24)(18,23)(20,21)(25,28)(26,29)(27,35)(30,39)(31,36)(32,40)(33,38)(34,37);(1,4,6,2)(3,8,5,7)(9,12)(10,11)(13,20)(14,17)(15,19)(16,18)(21,22)(23,24)(25,27)(26,32)(28,34)(29,36)(30,37)(31,33)(35,39)(38,40);(1,5,6,3)(2,8,4,7)(9,13)(10,16)(11,18)(12,20)(14,23)(15,21)(17,24)(19,22)(25,29)(26,28)(27,36)(30,38)(31,35)(32,34)(33,39)(37,40)Polycyclic presentation:
2^9|a^2=gh,c^2=gh,d^2=gh,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10318929);