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 | #45766 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4174 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,5)(3,7,8,4)(10,15)(13,16)(17,22)(21,23)(25,26)(27,31)(28,34)(29,33)(30,32)(35,37)(36,40)(38,39);(1,3)(2,4)(5,7)(6,8)(12,20)(17,22)(19,24)(21,23)(26,27)(29,37)(32,34)(36,38);(9,10)(11,16)(12,17)(13,18)(14,15)(19,23)(20,22)(21,24)(25,27,31,26)(28,34,30,32)(29,35,37,33)(36,40,38,39);(1,4)(2,3)(5,8)(6,7)(9,11,14,18)(10,16,15,13)(12,19,20,24)(17,23,22,21)(25,28,31,30)(26,32,27,34)(29,36,37,38)(33,39,35,40);(1,5,6,2)(3,7,8,4)(9,12)(10,17)(11,19)(13,21)(14,20)(15,22)(16,23)(18,24)(25,29,31,37)(26,33,27,35)(28,36,30,38)(32,39,34,40);(1,3)(2,7)(4,5)(6,8)(9,13,14,16)(10,18,15,11)(12,21,20,23)(17,24,22,19)(25,30,31,28)(26,34,27,32)(29,38,37,36)(33,40,35,39)Polycyclic presentation:
2^9|a^2=gi,c^2=i,d^2=h,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,e^d=egi,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, 9143648);