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 | |
4653 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,7,8,4)(9,10)(11,18)(12,16)(13,21)(14,15)(17,22)(19,20)(23,24)(25,26)(27,36)(28,30)(29,33)(31,35)(32,37)(34,40)(38,39);(2,6)(4,7)(10,15)(12,18)(17,21)(20,24)(27,37)(29,39)(32,36)(33,38);(1,2)(3,4)(5,6)(7,8)(10,15)(11,16)(13,22)(20,24)(25,27,31,37)(26,32,35,36)(28,39,34,29)(30,33,40,38);(1,3)(2,4)(5,8)(6,7)(9,11)(10,12)(13,19)(14,16)(15,18)(17,20)(21,24)(22,23)(25,28,31,34)(26,30,35,40)(27,29,37,39)(32,38,36,33);(1,4)(2,3)(5,7)(6,8)(9,12)(10,16)(11,15)(13,20)(14,18)(17,23)(19,21)(22,24)(25,29)(26,33)(27,28)(30,36)(31,39)(32,40)(34,37)(35,38);(1,3)(2,4)(5,8)(6,7)(9,13)(10,17)(11,19)(12,20)(14,22)(15,21)(16,23)(18,24)(25,30)(26,34)(27,38)(28,35)(29,36)(31,40)(32,39)(33,37)Polycyclic presentation:
2^9|a^2=ghi,c^2=gh,d^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8812392);