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 | |
4159 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,15)(12,19)(13,14)(16,21)(17,24)(18,22)(20,23)(25,26,31,30)(27,33,28,32)(29,34,37,38)(35,40,36,39);(1,2,5,7)(3,8,4,6)(9,10)(11,15)(12,16)(13,14)(17,23)(18,22)(19,21)(20,24)(25,27,31,28)(26,32,30,33)(29,36,37,35)(34,40,38,39);(1,3,5,4)(2,6,7,8)(10,14)(15,22)(16,19)(23,24);(9,10)(11,15)(12,16)(13,14)(17,23)(18,22)(19,21)(20,24)(25,28,31,27)(26,33,30,32)(29,36,37,35)(34,40,38,39);(1,2)(3,6)(4,8)(5,7)(9,11,13,18)(10,15,14,22)(12,20,21,17)(16,24,19,23)(25,29,31,37)(26,34,30,38)(27,35,28,36)(32,39,33,40);(1,4,5,3)(2,6,7,8)(9,12)(10,16)(11,17)(13,21)(14,19)(15,23)(18,20)(22,24)(25,30,31,26)(27,33,28,32)(29,34,37,38)(35,39,36,40)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=hi,d^2=g,e^2=gh,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9796677);