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 | |
4341 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10)(11,13)(12,19)(14,22)(15,16)(17,24)(18,23)(20,21)(25,26,31,35)(27,36,38,32)(28,30,34,39)(29,33,37,40);(2,3)(6,7)(9,11,15,12)(10,13,16,19)(14,18,24,20)(17,21,22,23)(25,27)(26,32)(28,37)(29,34)(30,33)(31,38)(35,36)(39,40);(1,3,5,2)(4,7,8,6)(10,16)(11,12)(17,22)(18,20);(1,4,5,8)(2,6,3,7)(9,12,15,11)(10,13,16,19)(14,20,24,18)(17,21,22,23)(25,28,31,34)(26,30,35,39)(27,37,38,29)(32,33,36,40);(9,13)(10,11)(12,16)(14,23)(15,19)(17,20)(18,22)(21,24)(25,29,31,37)(26,33,35,40)(27,34,38,28)(30,36,39,32);(9,14,15,24)(10,17,16,22)(11,18,12,20)(13,21,19,23)(25,30)(26,34)(27,33)(28,35)(29,32)(31,39)(36,37)(38,40)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=hi,d^2=gh,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=di,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, 9633662);