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 |
| prime radical | #43530 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4339 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,16)(11,13,18,23)(12,17,21,24)(14,22,20,19)(25,26,31,33)(27,36,38,32)(28,39,35,30)(29,34,40,37);(1,2)(3,8)(4,6)(5,7)(9,11)(10,13)(12,20)(14,21)(15,18)(16,23)(17,19)(22,24)(25,27,31,38)(26,32,33,36)(28,37,35,34)(29,39,40,30);(1,3,5,4)(2,6,7,8)(10,16)(11,18)(12,21)(19,22)(26,33)(28,35)(32,36)(34,37);(1,2)(3,6)(4,8)(5,7)(9,12,15,21)(10,17,16,24)(11,20,18,14)(13,19,23,22)(25,28,31,35)(26,30,33,39)(27,37,38,34)(29,36,40,32);(1,4,5,3)(2,6,7,8)(9,13)(10,18)(11,16)(12,22)(14,24)(15,23)(17,20)(19,21)(25,29,31,40)(26,34,33,37)(27,30,38,39)(28,32,35,36);(1,2)(3,6)(4,8)(5,7)(9,14,15,20)(10,19,16,22)(11,21,18,12)(13,17,23,24)(25,30)(26,35)(27,29)(28,33)(31,39)(32,34)(36,37)(38,40)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=ghi,d^2=h,e^2=hi,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, 10065454);