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 | |
4437 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,7)(5,8)(11,17)(15,18)(19,23)(22,24)(25,26)(27,33)(28,36)(29,30)(31,35)(32,37)(34,39)(38,40);(1,2,4,6)(3,5,7,8)(9,10)(11,15)(12,20)(13,14)(16,21)(17,18)(19,24)(22,23)(26,29)(28,37)(31,33)(34,38);(1,3)(2,5)(4,7)(6,8)(10,14)(15,18)(16,20)(22,24)(25,27,30,35)(26,31,29,33)(28,34,37,38)(32,39,36,40);(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,21)(22,23)(25,27)(26,31)(28,38)(29,33)(30,35)(32,40)(34,37)(36,39);(1,2)(3,5)(4,6)(7,8)(9,11)(10,15)(12,19)(13,17)(14,18)(16,22)(20,24)(21,23)(25,28)(26,32)(27,34)(29,36)(30,37)(31,39)(33,40)(35,38);(1,3)(2,5)(4,7)(6,8)(9,12,13,21)(10,16,14,20)(11,19,17,23)(15,22,18,24)(25,29,30,26)(27,33,35,31)(28,36,37,32)(34,40,38,39)Polycyclic presentation:
2^9|b^2=gh,c^2=g,f^2=gi,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, 8782131);