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 | |
4551 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(3,4)(9,10,14,13)(11,16,18,15)(12,17,20,21)(19,23,24,22)(25,26)(27,36)(28,34)(29,33)(30,32)(31,35)(37,39)(38,40);(1,2,6,5)(3,8,4,7)(11,18)(12,20)(15,16)(17,21)(25,27,31,30)(26,32,35,36)(28,37,29,38)(33,39,34,40);(1,3,6,4)(2,7,5,8)(9,11)(10,15)(12,19)(13,16)(14,18)(17,22)(20,24)(21,23)(25,28,31,29)(26,33,35,34)(27,37,30,38)(32,39,36,40);(1,4,6,3)(2,8,5,7)(9,11)(10,16)(12,19)(13,15)(14,18)(17,23)(20,24)(21,22)(25,29,31,28)(26,34,35,33)(27,37,30,38)(32,39,36,40);(1,5,6,2)(3,8,4,7)(9,12)(10,17)(11,19)(13,21)(14,20)(15,22)(16,23)(18,24)(25,26)(27,36)(28,33)(29,34)(30,32)(31,35)(37,40)(38,39);(9,13,14,10)(11,16,18,15)(12,21,20,17)(19,23,24,22)(25,30,31,27)(26,32,35,36)(28,38,29,37)(33,39,34,40)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=g,d^2=g,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8413136);