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 | #53476 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4243 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,8,6,4)(9,10)(11,17)(12,19)(13,21)(14,16)(15,22)(18,20)(23,24)(25,26)(27,28)(29,32)(30,37)(31,33)(34,39)(35,40)(36,38);(1,3)(2,4)(5,6)(7,8)(9,11)(10,12)(13,18)(14,19)(15,20)(16,17)(21,23)(22,24)(25,27,32,31)(26,33,29,28)(30,35,39,38)(34,40,37,36);(1,4)(2,3)(5,8)(6,7)(9,10,14,16)(11,12,19,17)(13,15,22,21)(18,20,24,23)(25,28)(26,31)(27,29)(30,36)(32,33)(34,38)(35,37)(39,40);(1,4,5,8)(2,6,7,3)(9,12,14,17)(10,11,16,19)(13,20,22,23)(15,18,21,24)(25,29,32,26)(27,33,31,28)(30,37,39,34)(35,40,38,36);(1,4)(2,3)(5,8)(6,7)(9,11)(10,12)(13,18)(14,19)(15,20)(16,17)(21,23)(22,24)(25,30)(26,34)(27,35)(28,36)(29,37)(31,38)(32,39)(33,40);(1,3)(2,4)(5,6)(7,8)(9,13,14,22)(10,15,16,21)(11,18,19,24)(12,20,17,23)(25,31,32,27)(26,33,29,28)(30,38,39,35)(34,40,37,36)Polycyclic presentation:
2^9|a^2=gh,b^2=i,c^2=hi,d^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8066122);