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 | |
4486 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,15)(12,20)(13,14)(16,22)(17,19)(18,24)(21,23)(25,26)(27,35)(28,30)(29,33)(31,32)(34,38)(36,37)(39,40);(1,2,5,4)(3,6,7,8)(9,10)(11,17)(12,16)(13,14)(15,19)(18,24)(20,22)(21,23)(26,32)(28,38)(33,35)(36,39);(2,4)(6,8)(10,14)(15,17)(16,20)(23,24)(25,27,31,29)(26,33,32,35)(28,36,38,39)(30,40,34,37);(3,7)(6,8)(9,11)(10,15)(12,21)(13,19)(14,17)(16,24)(18,22)(20,23)(25,28)(26,34)(27,36)(29,39)(30,32)(31,38)(33,37)(35,40);(1,3)(2,6)(4,8)(5,7)(9,12)(10,16)(11,18)(13,22)(14,20)(15,23)(17,24)(19,21)(25,29,31,27)(26,35,32,33)(28,36,38,39)(30,40,34,37);(1,4,5,2)(3,8,7,6)(9,10)(11,15)(12,16)(13,14)(17,19)(18,23)(20,22)(21,24)(25,30,31,34)(26,28,32,38)(27,37,29,40)(33,39,35,36)Polycyclic presentation:
2^9|b^2=ghi,c^2=g,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8232864);