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 | #52010 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4241 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)(3,4)(5,7)(6,8)(10,15)(11,19)(13,22)(20,23);(1,3)(2,6)(4,7)(5,8)(9,10)(11,16)(12,22)(13,18)(14,15)(17,19)(20,24)(21,23)(25,26,30,33)(27,28,35,31)(29,32,38,40)(34,36,37,39);(9,11,14,19)(10,16,15,17)(12,20,18,23)(13,21,22,24)(25,27)(26,31)(28,33)(29,34)(30,35)(32,39)(36,40)(37,38);(1,2,5,7)(3,4,8,6)(9,11)(10,17)(12,23)(13,21)(14,19)(15,16)(18,20)(22,24)(25,28,30,31)(26,27,33,35)(29,36,38,39)(32,34,40,37);(1,4,5,6)(2,3,7,8)(9,12)(10,13)(11,20)(14,18)(15,22)(16,21)(17,24)(19,23)(25,27)(26,28)(29,37)(30,35)(31,33)(32,39)(34,38)(36,40);(1,3)(2,4)(5,8)(6,7)(9,13)(10,18)(11,21)(12,15)(14,22)(16,23)(17,20)(19,24)(25,29)(26,32)(27,34)(28,36)(30,38)(31,39)(33,40)(35,37)Polycyclic presentation:
2^9|b^2=i,c^2=h,d^2=ghi,e^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10154416);