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 | C2×#121 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4129 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,5)(6,8)(9,10,14,17)(11,15,18,13)(12,16,20,23)(19,21,24,22)(25,26)(27,31)(28,34)(29,33)(30,32)(35,37)(36,40)(38,39);(3,8)(6,7)(9,11)(10,15)(12,19)(13,17)(14,18)(16,21)(20,24)(22,23)(26,27)(29,37)(32,34)(36,38);(1,3,5,8)(2,6,4,7)(11,18)(12,20)(13,15)(16,23)(25,27,31,26)(28,34,30,32)(29,35,37,33)(36,40,38,39);(3,8)(6,7)(9,12)(10,16)(11,19)(13,22)(14,20)(15,21)(17,23)(18,24)(25,28,31,30)(26,32,27,34)(29,36,37,38)(33,39,35,40);(1,3)(2,7)(4,6)(5,8)(9,13,14,15)(10,11,17,18)(12,21,20,22)(16,24,23,19)(25,29,31,37)(26,33,27,35)(28,36,30,38)(32,39,34,40);(1,4,5,2)(3,6,8,7)(9,11)(10,15)(12,19)(13,17)(14,18)(16,21)(20,24)(22,23)(25,30,31,28)(26,34,27,32)(29,38,37,36)(33,40,35,39)Polycyclic presentation:
2^9|a^2=hi,c^2=h,d^2=g,e^2=ghi,f^2=h,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, 8909596);