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 | |
4374 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,4)(3,8,5,7)(9,10,14,18)(11,15,19,24)(12,16,22,21)(13,17,23,20)(25,26,31,34)(27,33,28,32)(29,39,40,30)(35,38,37,36);(11,19)(13,23)(15,24)(17,20)(25,27,31,28)(26,32,34,33)(29,37,40,35)(30,36,39,38);(1,3,6,5)(2,7,4,8)(9,11)(10,15)(12,17)(13,21)(14,19)(16,23)(18,24)(20,22)(25,28,31,27)(26,33,34,32)(29,37,40,35)(30,36,39,38);(1,3,6,5)(2,8,4,7)(9,11)(10,15)(12,17)(13,21)(14,19)(16,23)(18,24)(20,22)(25,29)(26,30)(27,35)(28,37)(31,40)(32,38)(33,36)(34,39);(1,4,6,2)(3,8,5,7)(9,12)(10,16)(11,20)(13,15)(14,22)(17,19)(18,21)(23,24)(25,30)(26,29)(27,36)(28,38)(31,39)(32,37)(33,35)(34,40);(1,5,6,3)(2,8,4,7)(9,13)(10,17)(11,21)(12,15)(14,23)(16,19)(18,20)(22,24)(25,29)(26,30)(27,35)(28,37)(31,40)(32,38)(33,36)(34,39)Polycyclic presentation:
2^9|a^2=hi,b^2=g,c^2=h,d^2=gh,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10235449);