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 | |
4612 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,4,7,8)(25,26,29,30)(27,33,35,31)(28,32,37,36)(34,39,40,38);(2,6)(3,7)(9,10,14,15)(11,16,21,24)(12,17,22,20)(13,18,23,19)(26,30)(31,33)(32,36)(38,39);(1,3)(2,4)(5,7)(6,8)(10,15)(12,22)(16,24)(18,19)(25,27,29,35)(26,31,30,33)(28,34,37,40)(32,38,36,39);(1,4)(2,3)(5,8)(6,7)(9,11)(10,16)(12,19)(13,17)(14,21)(15,24)(18,22)(20,23)(25,26)(27,31)(28,36)(29,30)(32,37)(33,35)(34,39)(38,40);(9,12)(10,17)(11,19)(13,16)(14,22)(15,20)(18,21)(23,24)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,39)(37,40);(9,13)(10,18)(11,20)(12,16)(14,23)(15,19)(17,21)(22,24)(25,28)(26,32)(27,34)(29,37)(30,36)(31,38)(33,39)(35,40)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8424792);