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 | #59414 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4486 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10,15,17)(11,20,21,16)(12,14,19,24)(13,23,22,18)(25,26)(27,32)(28,30)(29,36)(31,34)(33,39)(35,37)(38,40);(1,2)(3,7)(4,8)(5,6)(9,11)(10,16)(12,13)(14,18)(15,21)(17,20)(19,22)(23,24)(25,27)(26,32)(28,36)(29,30)(31,35)(33,38)(34,37)(39,40);(1,3,5,4)(2,7,6,8)(10,17)(11,21)(13,22)(14,24)(27,35)(29,39)(32,37)(33,36);(9,12)(10,14)(11,22)(13,21)(15,19)(16,23)(17,24)(18,20)(25,28,31,38)(26,30,34,40)(27,33,35,36)(29,32,39,37);(1,4,5,3)(2,8,6,7)(9,13,15,22)(10,18,17,23)(11,12,21,19)(14,20,24,16)(25,29)(26,33)(27,30)(28,37)(31,39)(32,38)(34,36)(35,40);(9,14)(10,19)(11,18)(12,17)(13,20)(15,24)(16,22)(21,23)(25,30)(26,28)(27,29)(31,40)(32,36)(33,37)(34,38)(35,39)Polycyclic presentation:
2^9|a^2=h,c^2=gh,d^2=hi,e^2=g,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=di,e^c=ei,f^c=fh,e^d=ehi,f^d=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7995521);