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 | |
4571 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)(3,7)(4,6)(5,8)(9,10,14,17)(11,12,15,18)(13,16,22,24)(19,20,23,21)(25,26)(27,34)(28,33)(29,31)(30,32)(35,40)(36,38)(37,39);(1,3,5,4)(2,6,8,7)(9,11)(10,12)(13,19)(14,15)(16,20)(17,18)(21,24)(22,23)(25,27,30,29)(26,31,32,34)(28,36,37,35)(33,40,39,38);(1,3,5,4)(2,7,8,6)(11,15)(12,18)(19,23)(20,21)(26,32)(28,37)(31,34)(35,36);(1,2)(3,7)(4,6)(5,8)(9,10)(11,18)(12,15)(13,16)(14,17)(19,21)(20,23)(22,24)(25,28)(26,33)(27,35)(29,36)(30,37)(31,38)(32,39)(34,40);(1,2)(3,6)(4,7)(5,8)(9,12,14,18)(10,15,17,11)(13,21,22,20)(16,19,24,23)(25,26)(27,34)(28,33)(29,31)(30,32)(35,40)(36,38)(37,39);(1,4,5,3)(2,6,8,7)(9,13)(10,16)(11,19)(12,20)(14,22)(15,23)(17,24)(18,21)(25,29,30,27)(26,34,32,31)(28,36,37,35)(33,40,39,38)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=gh,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9962786);