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 | |
4350 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,4,7,8)(9,10)(11,15)(12,19)(13,14)(16,20)(17,21)(18,24)(22,23)(25,26)(27,36)(28,34)(29,33)(30,32)(31,35)(37,39)(38,40);(2,6)(3,7)(11,17)(15,21)(18,23)(22,24)(25,27,31,30)(26,32,35,36)(28,37,29,38)(33,39,34,40);(1,3)(2,4)(5,7)(6,8)(10,14)(15,21)(16,19)(22,24)(25,28,31,29)(26,33,35,34)(27,37,30,38)(32,39,36,40);(1,4)(2,3)(5,8)(6,7)(9,10)(11,15)(12,16)(13,14)(17,21)(18,22)(19,20)(23,24)(25,29,31,28)(26,34,35,33)(27,37,30,38)(32,39,36,40);(9,11,13,17)(10,15,14,21)(12,18,20,23)(16,22,19,24)(25,30,31,27)(26,32,35,36)(28,38,29,37)(33,39,34,40);(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,26)(27,32)(28,33)(29,34)(30,36)(31,35)(37,39)(38,40)Polycyclic presentation:
2^9|a^2=i,b^2=gi,c^2=gi,d^2=gi,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dgi,e^b=eghi,d^c=dghi,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10227503);