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 | |
4374 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,18)(11,15,16,12)(13,21,24,17)(19,20,23,22)(26,32)(28,37)(33,35)(36,38);(1,2)(3,4)(5,7)(6,8)(9,11,14,16)(10,15,18,12)(13,19,24,23)(17,22,21,20)(25,26,31,32)(27,33,29,35)(28,34,37,39)(30,36,40,38);(1,3)(2,6)(4,7)(5,8)(9,12)(10,16)(11,18)(13,20)(14,15)(17,23)(19,21)(22,24)(25,27,31,29)(26,33,32,35)(28,36,37,38)(30,34,40,39);(1,4,5,6)(2,3,7,8)(9,12)(10,11)(13,22)(14,15)(16,18)(17,23)(19,21)(20,24)(25,28)(26,34)(27,36)(29,38)(30,35)(31,37)(32,39)(33,40);(1,3)(2,4)(5,8)(6,7)(9,11)(10,15)(12,18)(13,23)(14,16)(17,20)(19,24)(21,22)(25,29,31,27)(26,35,32,33)(28,36,37,38)(30,34,40,39);(1,2)(3,6)(4,8)(5,7)(9,13)(10,17)(11,19)(12,20)(14,24)(15,22)(16,23)(18,21)(25,30,31,40)(26,36,32,38)(27,34,29,39)(28,33,37,35)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=g,d^2=hi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9130757);