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 | |
4820 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)(4,5)(6,8)(9,10)(11,16)(12,21)(13,17)(14,15)(18,19)(20,23)(22,24)(25,26)(27,30)(28,31)(29,32)(33,35)(34,37)(36,38)(39,40);(2,5)(6,7)(10,15)(12,17)(16,23)(18,22)(25,26,29,32)(27,33,35,30)(28,31,36,38)(34,39,40,37);(1,2)(3,6)(4,5)(7,8)(10,15)(13,21)(16,23)(19,24)(28,36)(31,38)(34,40)(37,39);(1,3,4,8)(2,6,5,7)(9,11,14,20)(10,16,15,23)(12,22,17,18)(13,19,21,24)(25,27,29,35)(26,30,32,33)(28,34,36,40)(31,37,38,39);(9,12,14,17)(10,13,15,21)(11,18,20,22)(16,24,23,19)(25,26)(27,33)(28,31)(29,32)(30,35)(34,39)(36,38)(37,40);(9,13)(10,17)(11,19)(12,15)(14,21)(16,18)(20,24)(22,23)(25,28)(26,31)(27,34)(29,36)(30,37)(32,38)(33,39)(35,40)Polycyclic presentation:
2^9|b^2=ghi,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8314415);