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 | |
4336 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)(11,18)(12,20)(13,15)(14,17)(16,22)(19,23)(21,24)(25,26,30,31)(27,33,34,28)(29,37,38,32)(35,36,40,39);(1,2)(3,8)(4,6)(5,7)(9,11,14,13)(10,15,17,18)(12,19,22,21)(16,23,20,24)(25,27,30,34)(26,28,31,33)(29,35,38,40)(32,36,37,39);(1,3,5,6)(2,4,7,8)(26,31)(27,34)(32,37)(35,40);(9,12)(10,16)(11,19)(13,21)(14,22)(15,23)(17,20)(18,24)(25,26)(27,28)(29,32)(30,31)(33,34)(35,36)(37,38)(39,40);(1,2)(3,4)(5,7)(6,8)(9,13,14,11)(10,15,17,18)(12,21,22,19)(16,23,20,24)(25,28)(26,27)(29,36)(30,33)(31,34)(32,35)(37,40)(38,39);(1,4)(2,6)(3,7)(5,8)(9,10)(11,15)(12,16)(13,18)(14,17)(19,23)(20,22)(21,24)(25,29,30,38)(26,32,31,37)(27,35,34,40)(28,36,33,39)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=hi,e^2=gh,f^2=h,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, 9591288);