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 | |
4449 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)(3,8)(4,6)(5,7)(9,10)(11,15)(12,18)(13,21)(14,16)(17,20)(19,24)(22,23)(25,26,31,33)(27,35,30,32)(28,34,38,29)(36,40,39,37);(1,3,5,4)(2,6,7,8)(9,11)(10,15)(12,19)(13,22)(14,20)(16,17)(18,24)(21,23)(25,27,31,30)(26,32,33,35)(28,36,38,39)(29,40,34,37);(10,16)(11,20)(18,22)(19,21)(26,33)(28,38)(32,35)(36,39);(9,11)(10,17)(12,21)(13,22)(14,20)(15,16)(18,24)(19,23)(25,28)(26,34)(27,36)(29,33)(30,39)(31,38)(32,37)(35,40);(1,4,5,3)(2,6,7,8)(9,12)(10,18)(11,21)(13,15)(14,23)(16,22)(17,24)(19,20)(25,29,31,34)(26,28,33,38)(27,37,30,40)(32,39,35,36);(1,2)(3,6)(4,8)(5,7)(9,13,14,24)(10,19,16,21)(11,22,20,18)(12,17,23,15)(25,30,31,27)(26,35,33,32)(28,39,38,36)(29,37,34,40)Polycyclic presentation:
2^9|a^2=hi,b^2=g,e^2=g,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, 9625770);