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 | |
4571 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,6,8,4)(9,10)(11,18)(12,16)(13,21)(14,15)(17,23)(19,20)(22,24)(25,26,31,35)(27,34,29,32)(28,33,39,30)(36,38,37,40);(1,3,5,8)(2,6,7,4)(9,11,15,16)(10,12,14,18)(13,19,23,24)(17,20,21,22)(25,27)(26,32)(28,36)(29,31)(30,38)(33,40)(34,35)(37,39);(1,3)(2,6)(4,7)(5,8)(10,14)(12,18)(17,21)(20,22)(25,28,31,39)(26,33,35,30)(27,36,29,37)(32,40,34,38);(2,7)(3,8)(9,12,15,18)(10,16,14,11)(13,22,23,20)(17,19,21,24)(25,28)(26,33)(27,37)(29,36)(30,35)(31,39)(32,38)(34,40);(1,3)(2,4)(5,8)(6,7)(9,13)(10,17)(11,19)(12,20)(14,21)(15,23)(16,24)(18,22)(25,29,31,27)(26,34,35,32)(28,37,39,36)(30,40,33,38);(1,4)(2,3)(5,6)(7,8)(9,14,15,10)(11,18,16,12)(13,21,23,17)(19,22,24,20)(25,30,31,33)(26,28,35,39)(27,38,29,40)(32,36,34,37)Polycyclic presentation:
2^9|a^2=ghi,b^2=hi,c^2=gh,d^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8841801);