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 | |
4405 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)(3,6,4,8)(26,30)(31,32)(33,38)(39,40);(1,3,5,4)(2,6,7,8)(9,10)(11,17)(12,20)(13,14)(15,19)(16,21)(18,23)(22,24)(25,26,29,30)(27,31,35,32)(28,33,37,38)(34,39,36,40);(2,7)(3,4)(10,14)(12,21)(15,17)(18,22)(25,27,29,35)(26,31,30,32)(28,36,37,34)(33,40,38,39);(3,4)(6,8)(9,11)(10,15)(12,22)(13,19)(14,17)(16,24)(18,21)(20,23)(25,26)(27,31)(28,33)(29,30)(32,35)(34,39)(36,40)(37,38);(1,4,5,3)(2,8,7,6)(9,12)(10,16)(11,18)(13,21)(14,20)(15,23)(17,24)(19,22)(25,27)(26,32)(28,34)(29,35)(30,31)(33,40)(36,37)(38,39);(1,2)(3,6)(4,8)(5,7)(9,10)(11,15)(12,16)(13,14)(17,19)(18,23)(20,21)(22,24)(25,28,29,37)(26,33,30,38)(27,34,35,36)(31,39,32,40)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,e^c=egi,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, 8947885);