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 | |
4568 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)(11,16)(12,20)(13,17)(14,18)(15,21)(19,23)(22,24)(25,26)(27,34)(28,31)(29,36)(30,32)(33,38)(35,39)(37,40);(1,2)(3,6)(4,7)(5,8)(9,11)(10,13)(12,19)(14,17)(15,23)(16,18)(20,22)(21,24)(25,27,31,30)(26,32,28,34)(29,35,38,37)(33,39,36,40);(1,3)(2,5)(4,8)(6,7)(12,21)(15,20)(19,24)(22,23)(25,26,31,28)(27,34,30,32)(29,33,38,36)(35,40,37,39);(3,8)(5,6)(9,12)(10,15)(11,19)(13,23)(14,21)(16,22)(17,24)(18,20)(25,28,31,26)(27,32,30,34)(29,36,38,33)(35,39,37,40);(1,2)(3,5)(4,7)(6,8)(9,11)(10,16)(12,19)(13,18)(14,17)(15,22)(20,23)(21,24)(25,29)(26,33)(27,35)(28,36)(30,37)(31,38)(32,39)(34,40);(1,3)(2,6)(4,8)(5,7)(9,13,14,16)(10,17,18,11)(12,22,21,23)(15,19,20,24)(25,30,31,27)(26,32,28,34)(29,37,38,35)(33,39,36,40)Polycyclic presentation:
2^9|b^2=h,c^2=h,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8609830);