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 | #43212 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4278 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10,15,17)(11,13,12,16)(14,18,23,24)(19,21,20,22)(25,26,31,33)(27,32,30,34)(28,29,37,40)(35,36,38,39);(1,2,4,5)(3,6,7,8)(9,11,15,12)(10,16,17,13)(14,19,23,20)(18,22,24,21)(25,27,31,30)(26,32,33,34)(28,35,37,38)(29,36,40,39);(10,17)(11,12)(18,24)(19,20)(26,33)(28,37)(32,34)(35,38);(9,12,15,11)(10,13,17,16)(14,20,23,19)(18,21,24,22)(25,28)(26,29)(27,35)(30,38)(31,37)(32,36)(33,40)(34,39);(1,3,4,7)(2,6,5,8)(9,13)(10,11)(12,17)(14,22)(15,16)(18,20)(19,24)(21,23)(25,29)(26,28)(27,36)(30,39)(31,40)(32,35)(33,37)(34,38);(9,14)(10,18)(11,19)(12,20)(13,21)(15,23)(16,22)(17,24)(25,30,31,27)(26,34,33,32)(28,38,37,35)(29,39,40,36)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,d^2=i,e^2=gi,f^2=hi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7868962);