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 | |
4465 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)(3,8)(4,7)(5,6)(25,26,31,30)(27,33,28,32)(29,34,38,37)(35,40,36,39);(1,2,6,5)(3,7,4,8)(9,10,14,12)(11,15,18,19)(13,16,21,20)(17,22,23,24)(25,27,31,28)(26,32,30,33)(29,35,38,36)(34,39,37,40);(1,3)(2,7)(4,6)(5,8)(10,12)(13,21)(15,19)(17,23)(25,27,31,28)(26,33,30,32)(29,36,38,35)(34,39,37,40);(1,2,6,5)(3,7,4,8)(9,11,14,18)(10,15,12,19)(13,17,21,23)(16,22,20,24)(25,28,31,27)(26,33,30,32)(29,35,38,36)(34,39,37,40);(1,4,6,3)(2,7,5,8)(9,12,14,10)(11,15,18,19)(13,20,21,16)(17,22,23,24)(25,29,31,38)(26,34,30,37)(27,35,28,36)(32,39,33,40);(1,5,6,2)(3,7,4,8)(9,13)(10,16)(11,17)(12,20)(14,21)(15,22)(18,23)(19,24)(25,30,31,26)(27,33,28,32)(29,37,38,34)(35,40,36,39)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=gh,d^2=gi,e^2=gi,f^2=g,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, 8870839);