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 | |
4432 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)(11,13,17,15)(12,16,20,24)(19,22,21,23)(25,26,31,36)(27,37,38,32)(28,33,35,30)(29,34,39,40);(1,2,5,4)(3,7,8,6)(9,10)(11,15)(12,16)(13,17)(14,18)(19,23)(20,24)(21,22)(27,38)(29,39)(32,37)(34,40);(1,3,5,8)(2,6,4,7)(11,17)(12,20)(13,15)(16,24)(25,27,31,38)(26,32,36,37)(28,29,35,39)(30,34,33,40);(3,8)(6,7)(9,11,14,17)(10,15,18,13)(12,21,20,19)(16,22,24,23)(25,28,31,35)(26,33,36,30)(27,39,38,29)(32,34,37,40);(1,3)(2,7)(4,6)(5,8)(9,12,14,20)(10,16,18,24)(11,19,17,21)(13,22,15,23)(25,29,31,39)(26,34,36,40)(27,28,38,35)(30,37,33,32);(1,4,5,2)(3,6,8,7)(9,13,14,15)(10,17,18,11)(12,22,20,23)(16,21,24,19)(25,30,31,33)(26,35,36,28)(27,40,38,34)(29,37,39,32)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,c^2=i,d^2=gh,e^2=gh,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, 8935993);