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 | |
4421 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,18)(11,12,16,15)(13,17,21,24)(19,20,23,22)(25,26)(27,33)(28,32)(29,30)(31,34)(35,40)(36,39)(37,38);(1,2,5,4)(3,7,8,6)(9,10)(11,15)(12,16)(13,17)(14,18)(19,22)(20,23)(21,24)(27,34)(31,33)(35,36)(39,40);(1,3,5,8)(2,6,4,7)(11,16)(12,15)(13,21)(17,24)(25,27,30,34)(26,31,29,33)(28,36,38,35)(32,40,37,39);(3,8)(6,7)(9,11)(10,15)(12,18)(13,19)(14,16)(17,22)(20,24)(21,23)(25,28,30,38)(26,32,29,37)(27,35,34,36)(31,39,33,40);(1,3)(2,7)(4,6)(5,8)(9,12,14,15)(10,16,18,11)(13,20,21,22)(17,23,24,19)(25,27)(26,33)(28,36)(29,31)(30,34)(32,39)(35,38)(37,40);(1,4,5,2)(3,6,8,7)(9,13,14,21)(10,17,18,24)(11,19,16,23)(12,20,15,22)(25,29,30,26)(27,31,34,33)(28,37,38,32)(35,39,36,40)Polycyclic presentation:
2^9|a^2=g,b^2=hi,c^2=i,d^2=h,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8935309);