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 | |
4657 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,15)(11,13,17,23)(12,16,21,24)(18,22,19,20)(25,26)(27,32)(28,36)(29,33)(30,34)(31,38)(35,37)(39,40);(1,2)(3,8)(4,6)(5,7)(9,11)(10,13)(12,19)(14,17)(15,23)(16,20)(18,21)(22,24)(25,27)(26,29)(28,35)(30,33)(31,37)(32,34)(36,39)(38,40);(1,3,5,4)(2,6,7,8)(10,15)(11,17)(16,24)(18,19);(9,11)(10,13)(12,18)(14,17)(15,23)(16,22)(19,21)(20,24)(25,28,30,38)(26,31,34,36)(27,35,33,40)(29,37,32,39);(1,4,5,3)(2,6,7,8)(9,12,14,21)(10,16,15,24)(11,18,17,19)(13,22,23,20)(25,27)(26,32)(28,35)(29,34)(30,33)(31,39)(36,37)(38,40);(1,2)(3,6)(4,8)(5,7)(9,13)(10,11)(12,20)(14,23)(15,17)(16,19)(18,24)(21,22)(25,29,30,32)(26,33,34,27)(28,37,38,39)(31,40,36,35)Polycyclic presentation:
2^9|a^2=h,c^2=hi,d^2=g,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 9613679);