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 | |
4408 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)(3,7)(9,10,16,14)(11,20,15,17)(12,19,13,18)(21,23,22,24)(25,26,31,35)(27,36,40,32)(28,29,39,33)(30,34,37,38);(1,2,4,5)(3,6,7,8)(9,11,16,15)(10,17,14,20)(12,21,13,22)(18,23,19,24)(25,27)(26,32)(28,37)(29,34)(30,39)(31,40)(33,38)(35,36);(1,2)(3,6)(4,5)(7,8)(9,12,16,13)(10,18,14,19)(11,21,15,22)(17,23,20,24)(25,26,31,35)(27,32,40,36)(28,33,39,29)(30,34,37,38);(1,2)(3,6)(4,5)(7,8)(9,13,16,12)(10,19,14,18)(11,21,15,22)(17,23,20,24)(25,28)(26,33)(27,37)(29,35)(30,40)(31,39)(32,38)(34,36);(1,3,4,7)(2,6,5,8)(9,14,16,10)(11,17,15,20)(12,19,13,18)(21,23,22,24)(25,29,31,33)(26,28,35,39)(27,38,40,34)(30,36,37,32);(9,15,16,11)(10,17,14,20)(12,22,13,21)(18,23,19,24)(25,30,31,37)(26,34,35,38)(27,39,40,28)(29,36,33,32)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=gi,d^2=hi,e^2=g,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, 8412215);