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 | |
4784 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)(3,7)(4,6)(5,8)(9,10)(11,17)(12,20)(13,14)(15,18)(16,21)(19,22)(23,24)(25,26)(27,30)(28,31)(29,32)(33,35)(34,38)(36,39)(37,40);(1,2,4,6)(3,5,8,7)(9,10)(11,15)(12,16)(13,14)(17,18)(19,22)(20,21)(23,24)(25,26,29,32)(27,30,35,33)(28,31,37,40)(34,38,36,39);(1,2)(3,5)(4,6)(7,8)(10,14)(11,18)(12,21)(22,23)(25,27,29,35)(26,30,32,33)(28,36,37,34)(31,39,40,38);(1,3)(2,5)(4,8)(6,7)(9,11,13,18)(10,15,14,17)(12,19,21,24)(16,22,20,23)(25,27)(26,30)(28,34)(29,35)(31,38)(32,33)(36,37)(39,40);(9,10)(11,17)(12,16)(13,14)(15,18)(19,23)(20,21)(22,24)(25,26)(27,33)(28,31)(29,32)(30,35)(34,39)(36,38)(37,40);(9,12,13,21)(10,16,14,20)(11,19,18,24)(15,22,17,23)(25,28,29,37)(26,31,32,40)(27,34,35,36)(30,38,33,39)Polycyclic presentation:
2^9|b^2=g,c^2=ghi,d^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8964986);