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 | |
4398 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(10,15)(11,16)(17,22)(18,23);(3,7)(6,8)(9,10,14,15)(11,12,16,19)(13,17,21,22)(18,20,23,24)(25,26,30,34)(27,32,37,31)(28,29,33,40)(35,39,38,36);(1,2)(3,6)(4,5)(7,8)(11,16)(12,19)(13,21)(17,22)(25,27,30,37)(26,31,34,32)(28,38,33,35)(29,39,40,36);(9,11)(10,12)(13,18)(14,16)(15,19)(17,20)(21,23)(22,24)(25,28,30,33)(26,29,34,40)(27,35,37,38)(31,36,32,39);(1,3,4,7)(2,6,5,8)(9,12)(10,16)(11,15)(13,20)(14,19)(17,23)(18,22)(21,24)(25,27)(26,32)(28,38)(29,36)(30,37)(31,34)(33,35)(39,40);(1,2)(3,6)(4,5)(7,8)(9,13)(10,17)(11,18)(12,20)(14,21)(15,22)(16,23)(19,24)(25,29)(26,33)(27,36)(28,34)(30,40)(31,38)(32,35)(37,39)Polycyclic presentation:
2^9|b^2=gi,c^2=i,d^2=i,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8901294);