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 | |
4302 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,8)(4,6)(5,7)(9,10)(11,20)(12,18)(13,19)(14,16)(15,17)(21,24)(22,23)(28,36)(32,39)(35,37)(38,40);(1,3,5,6)(2,4,7,8)(9,11,15,14)(10,16,17,20)(12,21,22,19)(13,23,24,18)(25,26)(27,34)(28,32)(29,31)(30,33)(35,40)(36,39)(37,38);(10,17)(12,22)(16,20)(19,21)(25,27,30,29)(26,31,33,34)(28,35,36,37)(32,38,39,40);(9,12,15,22)(10,18,17,23)(11,21,14,19)(13,20,24,16)(25,28)(26,32)(27,35)(29,37)(30,36)(31,38)(33,39)(34,40);(1,2)(3,4)(5,7)(6,8)(9,13)(10,19)(11,18)(12,20)(14,23)(15,24)(16,22)(17,21)(25,26)(27,31)(28,32)(29,34)(30,33)(35,38)(36,39)(37,40);(1,4)(2,6)(3,7)(5,8)(9,14,15,11)(10,20,17,16)(12,19,22,21)(13,18,24,23)(25,29,30,27)(26,31,33,34)(28,37,36,35)(32,38,39,40)Polycyclic presentation:
2^9|b^2=g,c^2=i,d^2=h,f^2=hi,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, 9620574);