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 | #59228 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4340 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(4,8)(6,7)(9,10)(11,18)(12,15)(13,20)(14,16)(17,21)(19,23)(22,24)(25,26)(27,31)(28,37)(29,39)(30,34)(32,38)(33,35)(36,40);(2,3)(6,7)(10,12)(13,21)(16,18)(19,22)(25,26,30,34)(27,31,36,40)(28,32,38,37)(29,33,35,39);(1,2,5,3)(4,6,8,7)(9,11,15,14)(10,16,12,18)(13,19,21,22)(17,23,20,24)(25,27,30,36)(26,31,34,40)(28,35,38,29)(32,39,37,33);(9,12,15,10)(11,18,14,16)(13,20,21,17)(19,24,22,23)(25,28)(26,32)(27,35)(29,36)(30,38)(31,39)(33,40)(34,37);(1,3,5,2)(4,7,8,6)(9,13)(10,17)(11,19)(12,20)(14,22)(15,21)(16,23)(18,24)(25,26)(27,31)(28,32)(29,33)(30,34)(35,39)(36,40)(37,38);(1,4)(2,6)(3,7)(5,8)(9,14,15,11)(10,18,12,16)(13,22,21,19)(17,24,20,23)(25,29,30,35)(26,33,34,39)(27,28,36,38)(31,32,40,37)Polycyclic presentation:
2^9|b^2=gh,c^2=gi,d^2=h,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8017161);