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 | #59048 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4247 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,5)(6,8)(9,10)(11,14)(12,17)(13,16)(15,20)(18,22)(19,24)(21,23)(25,26,31,30)(27,32,38,37)(28,34,29,33)(35,40,36,39);(1,3)(2,6)(4,7)(5,8)(9,10)(11,14)(12,15)(13,16)(17,20)(18,22)(19,23)(21,24)(25,27,31,38)(26,32,30,37)(28,35,29,36)(33,39,34,40);(1,3)(2,6)(4,7)(5,8)(10,11)(15,17)(16,18)(23,24)(25,28,31,29)(26,33,30,34)(27,35,38,36)(32,39,37,40);(3,8)(6,7)(9,11,14,10)(12,17,20,15)(13,16,22,18)(19,23,21,24)(25,27)(26,32)(28,35)(29,36)(30,37)(31,38)(33,39)(34,40);(1,4,5,2)(3,7,8,6)(9,12,14,20)(10,15,11,17)(13,21,22,19)(16,24,18,23)(25,29,31,28)(26,34,30,33)(27,36,38,35)(32,40,37,39);(1,3)(2,7)(4,6)(5,8)(9,13,14,22)(10,16,11,18)(12,19,20,21)(15,23,17,24)(25,30,31,26)(27,37,38,32)(28,34,29,33)(35,40,36,39)Polycyclic presentation:
2^9|a^2=ghi,b^2=ghi,c^2=ghi,d^2=i,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8653256);