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 | |
4485 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,8,4,6)(9,10)(11,19)(12,17)(13,21)(14,15)(16,20)(18,24)(22,23)(25,26)(27,36)(28,29)(30,40)(31,33)(32,37)(34,39)(35,38);(3,4)(6,8)(10,15)(12,23)(16,19)(18,21)(25,27,31,37)(26,32,33,36)(28,38,39,30)(29,40,34,35);(1,3,5,4)(2,6,7,8)(10,15)(11,20)(17,22)(18,21)(26,33)(27,37)(28,39)(35,40);(3,4)(6,8)(9,11,14,20)(10,16,15,19)(12,18,23,21)(13,22,24,17)(25,28)(26,34)(27,38)(29,33)(30,37)(31,39)(32,35)(36,40);(1,4,5,3)(2,6,7,8)(9,12)(10,17)(11,21)(13,19)(14,23)(15,22)(16,24)(18,20)(25,29,31,34)(26,28,33,39)(27,35,37,40)(30,36,38,32);(1,2)(3,6)(4,8)(5,7)(9,13,14,24)(10,18,15,21)(11,22,20,17)(12,16,23,19)(25,30,31,38)(26,35,33,40)(27,28,37,39)(29,36,34,32)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,c^2=gh,d^2=hi,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10079126);