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 | |
4325 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10)(11,14)(12,13)(15,16)(17,21)(18,24)(19,22)(20,23)(25,26)(27,37)(28,31)(29,34)(30,35)(32,36)(33,40)(38,39);(2,3)(6,7)(9,11,15,18)(10,14,16,24)(12,19,21,23)(13,22,17,20)(25,27)(26,33)(28,30)(29,38)(31,39)(32,40)(34,35)(36,37);(1,3,5,2)(4,7,8,6)(10,16)(11,18)(12,21)(20,22)(25,28,32,29)(26,31,36,34)(27,30,40,38)(33,39,37,35);(1,4,5,8)(2,6,3,7)(9,12)(10,17)(11,19)(13,16)(14,20)(15,21)(18,23)(22,24)(25,29,32,28)(26,34,36,31)(27,38,40,30)(33,35,37,39);(9,13,15,17)(10,12,16,21)(11,20,18,22)(14,23,24,19)(25,30,32,38)(26,35,36,39)(27,28,40,29)(31,33,34,37);(9,14)(10,18)(11,16)(12,20)(13,19)(15,24)(17,23)(21,22)(25,31,32,34)(26,28,36,29)(27,39,40,35)(30,37,38,33)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=g,d^2=g,e^2=gi,f^2=ghi,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, 9626355);