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 | |
4420 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,4)(3,7,5,8)(9,10,15,17)(11,16,12,18)(13,21,23,19)(14,22,24,20)(26,31)(27,35)(29,33)(36,38);(1,3,6,5)(2,7,4,8)(9,11)(10,16)(12,15)(13,20)(14,21)(17,18)(19,24)(22,23)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,37)(38,40);(1,3,6,5)(2,8,4,7)(10,17)(11,12)(19,21)(20,22)(27,35)(32,39)(36,38)(37,40);(1,4,6,2)(3,7,5,8)(9,12,15,11)(10,18,17,16)(13,20,23,22)(14,19,24,21)(25,27)(26,32)(28,38)(29,37)(30,35)(31,39)(33,40)(34,36);(1,5,6,3)(2,7,4,8)(9,13,15,23)(10,19,17,21)(11,20,12,22)(14,16,24,18)(25,28,30,34)(26,33,31,29)(27,36,35,38)(32,37,39,40);(9,14)(10,20)(11,21)(12,19)(13,16)(15,24)(17,22)(18,23)(25,29,30,33)(26,34,31,28)(27,37,35,40)(32,36,39,38)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=hi,d^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8065851);