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 | |
4358 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)(11,13)(12,21)(14,22)(15,17)(16,20)(18,19)(23,24)(25,26,31,35)(27,30,29,33)(28,34,39,32)(36,38,37,40);(1,2,5,4)(3,7,8,6)(9,11,15,20)(10,16,17,13)(12,18,22,24)(14,19,21,23)(25,27,31,29)(26,30,35,33)(28,37,39,36)(32,38,34,40);(1,3,5,8)(2,6,4,7)(9,12,15,22)(10,14,17,21)(11,18,20,24)(13,23,16,19)(25,28,31,39)(26,32,35,34)(27,36,29,37)(30,40,33,38);(3,8)(6,7)(9,12)(10,14)(11,18)(13,23)(15,22)(16,19)(17,21)(20,24)(25,29,31,27)(26,33,35,30)(28,36,39,37)(32,40,34,38);(1,3)(2,7)(4,6)(5,8)(9,13,15,16)(10,11,17,20)(12,19,22,23)(14,24,21,18)(25,28)(26,34)(27,36)(29,37)(30,38)(31,39)(32,35)(33,40);(1,4,5,2)(3,6,8,7)(9,14)(10,12)(11,19)(13,24)(15,21)(16,18)(17,22)(20,23)(25,30,31,33)(26,27,35,29)(28,38,39,40)(32,37,34,36)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=gi,d^2=gh,e^2=g,f^2=i,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, 8935227);