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 | |
4636 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10)(11,17)(12,20)(13,14)(15,18)(16,21)(19,22)(23,24)(25,26)(27,31)(28,35)(29,30)(32,37)(33,40)(34,38)(36,39);(2,3)(6,7)(10,14)(12,21)(15,17)(19,24)(26,30)(31,38)(32,35)(39,40);(1,3,5,2)(4,7,8,6)(10,14)(11,18)(16,20)(19,24)(25,27,29,34)(26,31,30,38)(28,36,37,33)(32,40,35,39);(1,4,5,8)(2,6,3,7)(9,11,13,18)(10,15,14,17)(12,19,21,24)(16,22,20,23)(25,28,29,37)(26,32,30,35)(27,33,34,36)(31,39,38,40);(9,10)(11,15)(12,16)(13,14)(17,18)(19,22)(20,21)(23,24)(25,27)(26,31)(28,33)(29,34)(30,38)(32,39)(35,40)(36,37);(9,12)(10,16)(11,19)(13,21)(14,20)(15,22)(17,23)(18,24)(25,26)(27,31)(28,32)(29,30)(33,39)(34,38)(35,37)(36,40)Polycyclic presentation:
2^9|a^2=i,c^2=gh,d^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8669141);