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 | |
4268 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,5)(3,8,7,4)(9,10)(11,12)(13,21)(14,18)(15,17)(16,23)(19,24)(20,22)(25,26)(27,32)(28,29)(30,39)(31,35)(33,38)(34,36)(37,40);(1,3,6,7)(2,4,5,8)(9,11)(10,12)(13,22)(14,15)(16,24)(17,18)(19,23)(20,21)(25,27,31,37)(26,32,35,40)(28,36,38,30)(29,34,33,39);(9,12)(10,15)(11,18)(13,20)(14,17)(16,22)(19,21)(23,24)(25,28)(26,33)(27,36)(29,35)(30,37)(31,38)(32,39)(34,40);(2,5)(4,8)(9,12)(10,11)(13,20)(14,17)(15,18)(16,19)(21,22)(23,24)(25,29,31,33)(26,28,35,38)(27,34,37,39)(30,32,36,40);(1,4,6,8)(2,7,5,3)(9,13,14,23)(10,16,18,21)(11,19,15,22)(12,20,17,24)(25,28)(26,29)(27,36)(30,37)(31,38)(32,34)(33,35)(39,40);(1,5,6,2)(3,8,7,4)(9,11)(10,17)(12,18)(13,19)(14,15)(16,24)(20,21)(22,23)(25,30,31,36)(26,34,35,39)(27,28,37,38)(29,32,33,40)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,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, 10358818);