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 | #2541 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4204 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)(9,10,15,14)(11,16,18,13)(12,17,21,22)(19,23,24,20)(25,26,29,30)(27,33,34,31)(28,36,37,32)(35,38,40,39);(1,2,5,4)(3,7,8,6)(9,11,15,18)(10,16,14,13)(12,19,21,24)(17,23,22,20)(26,30)(27,34)(32,36)(35,40);(1,3,5,8)(2,6,4,7)(11,18)(13,16)(19,24)(20,23)(25,27,29,34)(26,31,30,33)(28,35,37,40)(32,38,36,39);(1,4,5,2)(3,7,8,6)(9,12)(10,17)(11,19)(13,20)(14,22)(15,21)(16,23)(18,24)(25,26)(27,31)(28,32)(29,30)(33,34)(35,38)(36,37)(39,40);(9,13)(10,11)(12,20)(14,18)(15,16)(17,19)(21,23)(22,24)(25,28,29,37)(26,32,30,36)(27,35,34,40)(31,38,33,39);(9,14,15,10)(11,13,18,16)(12,17,21,22)(19,23,24,20)Polycyclic presentation:
2^9|a^2=ghi,b^2=hi,c^2=g,d^2=gh,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8458114);