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 | #41957 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4200 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,7)(11,14)(16,18)(19,21)(23,24)(25,26,30,28)(27,31,33,34)(29,32,37,36)(35,38,39,40);(1,2,5,4)(3,7,8,6)(9,10,15,12)(11,16,14,18)(13,20,22,17)(19,24,21,23)(27,33)(29,37)(31,34)(32,36);(1,3,5,8)(2,6,4,7)(9,11,15,14)(10,16,12,18)(13,19,22,21)(17,23,20,24)(25,27,30,33)(26,31,28,34)(29,35,37,39)(32,38,36,40);(1,4,5,2)(3,7,8,6)(9,12,15,10)(11,18,14,16)(13,20,22,17)(19,24,21,23)(25,26,30,28)(27,34,33,31)(29,32,37,36)(35,40,39,38);(9,13,15,22)(10,17,12,20)(11,19,14,21)(16,23,18,24)(25,28,30,26)(27,31,33,34)(29,36,37,32)(35,38,39,40);(9,14,15,11)(10,18,12,16)(13,21,22,19)(17,24,20,23)(25,29)(26,32)(27,35)(28,36)(30,37)(31,38)(33,39)(34,40)Polycyclic presentation:
2^9|a^2=gi,b^2=gh,c^2=hi,d^2=hi,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8714080);