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 | |
4482 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)(11,17)(15,18)(19,23)(22,24)(25,26)(27,29)(28,36)(30,31)(32,35)(33,39)(34,40)(37,38);(2,3)(4,8)(9,10)(11,15)(12,20)(13,14)(16,21)(17,18)(19,24)(22,23)(26,31)(29,32)(33,36)(38,40);(1,2,5,3)(4,6,8,7)(10,14)(15,18)(16,20)(22,24)(25,27,30,35)(26,29,31,32)(28,34,39,37)(33,38,36,40);(2,3)(6,7)(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,21)(22,23)(25,27)(26,32)(28,37)(29,31)(30,35)(33,38)(34,39)(36,40);(1,3,5,2)(4,7,8,6)(9,11,13,17)(10,15,14,18)(12,19,21,23)(16,22,20,24)(25,28)(26,33)(27,34)(29,38)(30,39)(31,36)(32,40)(35,37);(1,4)(2,6)(3,7)(5,8)(9,12)(10,16)(11,19)(13,21)(14,20)(15,22)(17,23)(18,24)(25,29,30,32)(26,27,31,35)(28,38,39,40)(33,34,36,37)Polycyclic presentation:
2^9|c^2=gh,e^2=i,f^2=g,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, 8761457);