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 | |
4509 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)(10,16)(11,12)(17,23)(18,20)(25,26)(27,37)(28,33)(29,34)(30,35)(31,36)(32,38)(39,40);(1,3,5,4)(2,6,7,8)(9,10,15,16)(11,14,12,19)(13,17,22,23)(18,21,20,24)(27,38)(28,39)(32,37)(33,40);(9,11,15,12)(10,14,16,19)(13,18,22,20)(17,21,23,24)(25,27)(26,32)(28,35)(29,39)(30,40)(31,38)(33,34)(36,37);(2,7)(6,8)(9,12,15,11)(10,14,16,19)(13,20,22,18)(17,21,23,24)(25,28,31,39)(26,33,36,40)(27,35,38,29)(30,32,34,37);(1,4,5,3)(2,8,7,6)(9,13,15,22)(10,17,16,23)(11,18,12,20)(14,21,19,24)(25,29,31,35)(26,34,36,30)(27,39,38,28)(32,33,37,40);(1,2)(3,6)(4,8)(5,7)(9,14)(10,12)(11,16)(13,21)(15,19)(17,20)(18,23)(22,24)(25,30,31,34)(26,35,36,29)(27,40,38,33)(28,37,39,32)Polycyclic presentation:
2^9|b^2=g,c^2=hi,d^2=h,e^2=gi,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8371406);