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 | |
4588 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,8)(9,10)(11,13)(12,20)(14,17)(15,19)(16,21)(18,24)(22,23)(25,26,31,35)(27,32,36,38)(28,33,37,40)(29,30,34,39);(3,7)(6,8)(9,11)(10,15)(12,18)(13,17)(14,19)(16,24)(20,22)(21,23)(25,27,31,36)(26,32,35,38)(28,30,37,39)(29,33,34,40);(1,2,5,4)(3,6,7,8)(12,21)(16,20)(18,23)(22,24)(27,36)(28,37)(32,38)(33,40);(1,3,5,7)(2,6,4,8)(9,12)(10,16)(11,18)(13,22)(14,21)(15,24)(17,20)(19,23)(25,28)(26,33)(27,30)(29,38)(31,37)(32,34)(35,40)(36,39);(1,4,5,2)(3,8,7,6)(9,13,14,15)(10,11,17,19)(12,22,21,24)(16,18,20,23)(25,29)(26,30)(27,33)(28,38)(31,34)(32,37)(35,39)(36,40);(9,11)(10,13)(12,23)(14,19)(15,17)(16,24)(18,21)(20,22)(25,30)(26,34)(27,37)(28,36)(29,35)(31,39)(32,40)(33,38)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=h,d^2=h,e^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8619912);