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 | |
4308 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,3)(4,8)(9,10,14,17)(11,12,15,19)(13,16,22,24)(18,20,21,23)(25,26)(27,36)(28,39)(29,30)(31,32)(33,37)(34,38)(35,40);(2,3)(4,8)(9,11)(10,12)(13,21)(14,15)(16,23)(17,19)(18,22)(20,24)(26,32)(27,37)(29,35)(34,39);(1,2,5,3)(4,6,8,7)(11,15)(12,19)(18,21)(20,23)(25,27)(26,33)(28,35)(29,38)(30,39)(31,37)(32,36)(34,40);(2,3)(6,7)(9,12,14,19)(10,15,17,11)(13,20,22,23)(16,21,24,18)(25,28,31,38)(26,34,32,39)(27,35,37,29)(30,33,40,36);(1,3,5,2)(4,7,8,6)(9,10)(11,12)(13,16)(14,17)(15,19)(18,20)(21,23)(22,24)(25,29,31,35)(26,30,32,40)(27,38,37,28)(33,39,36,34);(1,4,5,8)(2,6,3,7)(9,13)(10,16)(11,18)(12,20)(14,22)(15,21)(17,24)(19,23)(25,30)(26,35)(27,39)(28,33)(29,32)(31,40)(34,37)(36,38)Polycyclic presentation:
2^9|a^2=hi,c^2=gh,d^2=ghi,e^2=h,f^2=gh,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, 8748531);