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 | |
4336 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,17)(11,16,13,19)(12,18,23,22)(14,21,24,20)(25,26,30,28)(27,33,35,31)(29,36,38,32)(34,39,37,40);(1,2,5,4)(3,7,8,6)(9,11)(10,16)(12,21)(13,15)(14,22)(17,19)(18,24)(20,23)(26,28)(31,33)(32,36)(39,40);(1,3,5,8)(2,6,4,7)(10,17)(11,13)(12,23)(14,24)(25,27,30,35)(26,31,28,33)(29,37,38,34)(32,40,36,39);(3,8)(6,7)(9,12,15,23)(10,18,17,22)(11,21,13,20)(14,16,24,19)(25,27)(26,31)(28,33)(29,34)(30,35)(32,39)(36,40)(37,38);(1,3)(2,7)(4,6)(5,8)(9,13,15,11)(10,19,17,16)(12,21,23,20)(14,18,24,22)(25,28,30,26)(27,31,35,33)(29,36,38,32)(34,39,37,40);(1,4,5,2)(3,6,8,7)(9,14,15,24)(10,20,17,21)(11,22,13,18)(12,16,23,19)(25,29,30,38)(26,32,28,36)(27,34,35,37)(31,39,33,40)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,c^2=hi,d^2=gh,e^2=h,f^2=gi,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, 8936234);