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 | |
4458 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)(11,14)(12,15)(13,18)(16,21)(17,20)(19,23)(22,24)(25,26,31,36)(27,37,38,32)(28,33,35,30)(29,34,40,39);(1,2)(3,6)(4,7)(5,8)(9,10,14,11)(12,17,20,15)(13,16,21,18)(19,23,24,22)(27,38)(29,40)(32,37)(34,39);(1,3)(2,5)(4,8)(6,7)(10,11)(15,17)(16,18)(22,23)(25,27,31,38)(26,32,36,37)(28,40,35,29)(30,39,33,34);(1,2)(3,5)(4,7)(6,8)(9,11,14,10)(12,17,20,15)(13,18,21,16)(19,23,24,22)(25,28,31,35)(26,33,36,30)(27,29,38,40)(32,39,37,34);(1,3)(2,6)(4,8)(5,7)(9,12)(10,15)(11,17)(13,19)(14,20)(16,22)(18,23)(21,24)(25,29)(26,34)(27,28)(30,32)(31,40)(33,37)(35,38)(36,39);(1,2)(3,5)(4,7)(6,8)(9,13)(10,16)(11,18)(12,19)(14,21)(15,22)(17,23)(20,24)(25,30,31,33)(26,35,36,28)(27,39,38,34)(29,37,40,32)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=gh,d^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9996473);