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 | |
4434 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)(3,7)(4,6)(5,8)(9,10)(11,13)(12,17)(14,22)(15,16)(18,24)(19,23)(20,21)(25,26,30,33)(27,29,35,31)(28,32,36,39)(34,37,40,38);(1,2,4,6)(3,5,8,7)(9,11,15,12)(10,13,16,17)(14,19,24,20)(18,21,22,23)(28,36)(32,39)(34,40)(37,38);(1,2)(3,5)(4,6)(7,8)(10,16)(11,12)(18,22)(19,20)(25,27,30,35)(26,31,33,29)(28,34,36,40)(32,38,39,37);(1,3)(2,5)(4,8)(6,7)(9,12,15,11)(10,17,16,13)(14,20,24,19)(18,23,22,21)(25,27)(26,29)(28,34)(30,35)(31,33)(32,37)(36,40)(38,39);(9,13,15,17)(10,11,16,12)(14,23,24,21)(18,20,22,19)(25,28)(26,32)(27,34)(29,37)(30,36)(31,38)(33,39)(35,40);(9,14,15,24)(10,18,16,22)(11,19,12,20)(13,21,17,23)(25,29)(26,27)(28,37)(30,31)(32,34)(33,35)(36,38)(39,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=i,c^2=ghi,d^2=gi,e^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9816975);