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 | |
4391 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,7)(5,8)(9,10)(11,18)(12,16)(13,21)(14,15)(17,22)(19,20)(23,24)(27,33)(31,38)(34,37)(39,40);(1,2,4,6)(3,5,7,8)(9,11,14,16)(10,12,15,18)(13,19,22,23)(17,20,21,24)(25,26)(27,31)(28,35)(29,30)(32,36)(33,38)(34,40)(37,39);(1,3)(2,5)(4,7)(6,8)(10,15)(12,18)(13,22)(19,23)(26,30)(27,33)(28,36)(39,40);(9,10)(11,12)(13,17)(14,15)(16,18)(19,20)(21,22)(23,24)(25,27)(26,31)(28,37)(29,33)(30,38)(32,40)(34,36)(35,39);(1,2)(3,5)(4,6)(7,8)(9,12)(10,16)(11,15)(13,20)(14,18)(17,23)(19,21)(22,24)(25,28)(26,32)(27,34)(29,36)(30,35)(31,39)(33,37)(38,40);(1,3)(2,5)(4,7)(6,8)(9,13,14,22)(10,17,15,21)(11,19,16,23)(12,20,18,24)(25,26)(27,31)(28,32)(29,30)(33,38)(34,39)(35,36)(37,40)Polycyclic presentation:
2^9|b^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8948351);