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 | |
4455 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(10,15)(11,16)(17,22)(18,23)(25,26)(27,35)(28,33)(29,34)(30,31)(32,36)(37,40)(38,39);(1,2,5,4)(3,7,8,6)(9,10,14,15)(11,12,16,19)(13,17,21,22)(18,20,23,24)(27,36)(29,37)(32,35)(34,40);(1,3,5,8)(2,6,4,7)(11,16)(12,19)(13,21)(17,22)(25,27,31,36)(26,32,30,35)(28,29,39,37)(33,40,38,34);(3,8)(6,7)(9,11)(10,12)(13,18)(14,16)(15,19)(17,20)(21,23)(22,24)(25,28,31,39)(26,33,30,38)(27,37,36,29)(32,34,35,40);(1,3)(2,7)(4,6)(5,8)(9,12)(10,16)(11,15)(13,20)(14,19)(17,23)(18,22)(21,24)(25,29,31,37)(26,34,30,40)(27,28,36,39)(32,38,35,33);(1,4,5,2)(3,6,8,7)(9,13)(10,17)(11,18)(12,20)(14,21)(15,22)(16,23)(19,24)(25,30,31,26)(27,32,36,35)(28,38,39,33)(29,40,37,34)Polycyclic presentation:
2^9|b^2=ghi,c^2=i,d^2=h,e^2=h,f^2=i,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, 8935874);