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 | #35436 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4432 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)(9,10,15,17)(11,13,12,16)(14,22,23,18)(19,24,20,21)(25,26)(27,31)(28,32)(29,30)(33,35)(34,39)(36,40)(37,38);(1,2,6,8)(3,5,7,4)(9,11,15,12)(10,16,17,13)(14,19,23,20)(18,24,22,21)(25,27,30,33)(26,31,29,35)(28,34,38,36)(32,39,37,40);(1,3,6,7)(2,4,8,5)(10,17)(13,16)(14,23)(19,20)(26,29)(27,33)(32,37)(34,36);(1,4,6,5)(2,3,8,7)(9,12,15,11)(10,13,17,16)(14,20,23,19)(18,21,22,24)(25,28)(26,32)(27,34)(29,37)(30,38)(31,39)(33,36)(35,40);(1,2)(3,4)(5,7)(6,8)(9,13)(10,12)(11,17)(14,21)(15,16)(18,20)(19,22)(23,24)(25,27)(26,31)(28,36)(29,35)(30,33)(32,40)(34,38)(37,39);(1,5,6,4)(2,7,8,3)(9,14,15,23)(10,18,17,22)(11,19,12,20)(13,21,16,24)(25,29,30,26)(27,35,33,31)(28,37,38,32)(34,40,36,39)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=ghi,d^2=g,f^2=h,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, 8851658);