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 | #44206 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4348 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,8)(3,5,7,4)(9,10)(11,15)(12,18)(13,16)(14,17)(19,23)(20,24)(21,22)(25,26)(27,34)(28,31)(29,37)(30,33)(32,38)(35,39)(36,40);(1,3,6,7)(2,5,8,4)(11,12)(15,18)(19,20)(23,24)(25,27)(26,31)(28,30)(29,35)(32,39)(33,34)(36,38)(37,40);(1,4)(2,7)(3,8)(5,6)(9,11,14,12)(10,15,17,18)(13,19,22,20)(16,23,21,24)(25,26,30,33)(27,31,28,34)(29,32,38,37)(35,39,36,40);(1,2)(3,4)(5,7)(6,8)(9,12,14,11)(10,15,17,18)(13,20,22,19)(16,23,21,24)(25,26)(27,34)(28,31)(29,32)(30,33)(35,40)(36,39)(37,38);(1,5,6,4)(2,7,8,3)(9,10)(11,18)(12,15)(13,21)(14,17)(16,22)(19,23)(20,24)(25,28,30,27)(26,31,33,34)(29,35,38,36)(32,40,37,39);(1,2)(3,5)(4,7)(6,8)(9,13)(10,16)(11,19)(12,20)(14,22)(15,23)(17,21)(18,24)(25,29)(26,32)(27,35)(28,36)(30,38)(31,39)(33,37)(34,40)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=hi,d^2=ghi,e^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9954225);