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 | |
4447 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,18)(11,13,12,16)(14,22,23,17)(19,24,20,21)(25,26)(27,34)(28,33)(29,38)(30,35)(31,36)(32,39)(37,40);(1,2)(3,5)(4,6)(7,8)(9,11,15,12)(10,16,18,13)(14,19,23,20)(17,24,22,21)(25,27)(26,32)(28,35)(29,37)(30,38)(31,39)(33,40)(34,36);(1,2,4,6)(3,7,8,5)(9,11,15,12)(10,16,18,13)(14,20,23,19)(17,21,22,24)(25,28,31,29)(26,33,36,38)(27,35,39,37)(30,32,40,34);(3,8)(5,7)(9,12,15,11)(10,13,18,16)(14,20,23,19)(17,21,22,24)(25,29,31,28)(26,33,36,38)(27,37,39,35)(30,32,40,34);(1,3)(2,5)(4,8)(6,7)(9,13)(10,12)(11,18)(14,21)(15,16)(17,20)(19,22)(23,24)(25,27)(26,34)(28,37)(29,35)(30,38)(31,39)(32,36)(33,40);(1,2)(3,5)(4,6)(7,8)(9,14)(10,17)(11,19)(12,20)(13,21)(15,23)(16,24)(18,22)(25,30,31,40)(26,35,36,37)(27,38,39,33)(28,34,29,32)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=gi,d^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8850664);