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 | |
4765 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,3)(4,8)(9,10,14,15)(11,18,16,12)(13,21,22,17)(19,20,23,24)(27,33)(31,34)(35,39)(38,40);(2,3)(4,8)(10,15)(12,18)(13,22)(19,23)(25,26)(27,31)(28,36)(29,30)(32,37)(33,34)(35,40)(38,39);(1,2,5,3)(4,6,8,7)(10,15)(11,16)(17,21)(19,23)(26,30)(31,34)(32,36)(38,40);(2,3)(6,7)(9,11)(10,12)(13,19)(14,16)(15,18)(17,20)(21,24)(22,23)(25,26)(27,34)(28,32)(29,30)(31,33)(35,40)(36,37)(38,39);(1,3,5,2)(4,7,8,6)(9,12)(10,16)(11,15)(13,20)(14,18)(17,23)(19,21)(22,24)(25,27,29,33)(26,31,30,34)(28,35,37,39)(32,38,36,40);(1,4)(2,6)(3,7)(5,8)(9,13)(10,17)(11,19)(12,20)(14,22)(15,21)(16,23)(18,24)(25,28)(26,32)(27,35)(29,37)(30,36)(31,38)(33,39)(34,40)Polycyclic presentation:
2^9|a^2=g,c^2=gh,e^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8761426);