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 |
| prime radical | #34967 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4383 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)(3,5)(4,7)(6,8)(9,10)(11,13)(12,17)(14,19)(15,16)(18,23)(20,22)(21,24)(25,26,31,28)(27,32,35,37)(29,33,40,38)(30,34,36,39);(2,3)(6,7)(9,11,15,18)(10,13,16,23)(12,20,21,14)(17,22,24,19)(25,27)(26,32)(28,37)(29,30)(31,35)(33,34)(36,40)(38,39);(1,2,5,3)(4,6,8,7)(10,16)(11,18)(14,20)(17,24)(27,35)(30,36)(32,37)(34,39);(1,3,5,2)(4,7,8,6)(9,12)(10,17)(11,20)(13,22)(14,18)(15,21)(16,24)(19,23)(25,28,31,26)(27,32,35,37)(29,33,40,38)(30,39,36,34);(1,4,5,8)(2,6,3,7)(9,13,15,23)(10,18,16,11)(12,19,21,22)(14,17,20,24)(25,29)(26,33)(27,36)(28,38)(30,35)(31,40)(32,39)(34,37);(9,14,15,20)(10,19,16,22)(11,12,18,21)(13,17,23,24)(25,30,31,36)(26,34,28,39)(27,29,35,40)(32,33,37,38)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,c^2=g,d^2=ghi,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8309659);