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 | |
4262 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,7)(9,10,14,17)(11,15,12,19)(13,16,22,21)(18,23,20,24)(25,26)(27,30)(28,33)(29,32)(31,35)(34,40)(36,39)(37,38);(9,10)(11,15)(12,19)(13,16)(14,17)(18,23)(20,24)(21,22)(25,27,31,34)(26,30,35,40)(28,37,29,36)(32,39,33,38);(1,2,5,4)(3,6,8,7)(9,11,14,12)(10,15,17,19)(13,18,22,20)(16,23,21,24)(25,28,31,29)(26,32,35,33)(27,36,34,37)(30,38,40,39);(1,3,5,8)(2,6,4,7)(9,10)(11,15)(12,19)(13,21)(14,17)(16,22)(18,24)(20,23)(25,26)(27,30)(28,32)(29,33)(31,35)(34,40)(36,38)(37,39);(1,3)(2,6)(4,7)(5,8)(9,12,14,11)(10,15,17,19)(13,20,22,18)(16,23,21,24)(25,29,31,28)(26,33,35,32)(27,37,34,36)(30,39,40,38);(1,4,5,2)(3,7,8,6)(9,13,14,22)(10,16,17,21)(11,18,12,20)(15,23,19,24)(25,30)(26,34)(27,35)(28,38)(29,39)(31,40)(32,37)(33,36)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=gh,d^2=hi,e^2=gi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10358821);