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 | |
4368 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)(4,5)(6,8)(9,10)(11,17)(12,20)(13,14)(15,18)(16,21)(19,22)(23,24)(25,26,30,33)(27,31,34,29)(28,32,36,40)(35,39,37,38);(3,8)(6,7)(10,14)(15,17)(16,20)(22,23)(25,27)(26,31)(28,35)(29,33)(30,34)(32,39)(36,37)(38,40);(1,3,5,8)(2,6,4,7)(10,14)(11,18)(12,21)(22,23)(27,34)(28,36)(29,31)(32,40);(3,8)(6,7)(9,11,13,18)(10,15,14,17)(12,19,21,24)(16,22,20,23)(25,27)(26,31)(28,37)(29,33)(30,34)(32,38)(35,36)(39,40);(1,3)(2,7)(4,6)(5,8)(9,10)(11,17)(12,16)(13,14)(15,18)(19,23)(20,21)(22,24)(25,28)(26,32)(27,35)(29,38)(30,36)(31,39)(33,40)(34,37);(1,4,5,2)(3,6,8,7)(9,12,13,21)(10,16,14,20)(11,19,18,24)(15,22,17,23)(25,29,30,31)(26,27,33,34)(28,38,36,39)(32,35,40,37)Polycyclic presentation:
2^9|a^2=ghi,c^2=gh,d^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8927245);