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 | #50668 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4299 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,6,4,7)(9,10,15,17)(11,19,20,16)(12,18,21,24)(13,22,23,14)(25,26,31,28)(27,32,34,30)(29,33,39,36)(35,40,37,38);(1,2)(3,6)(4,7)(5,8)(9,11)(10,16)(12,13)(14,18)(15,20)(17,19)(21,23)(22,24)(25,27,31,34)(26,32,28,30)(29,37,39,35)(33,38,36,40);(1,3,5,4)(2,6,8,7)(10,17)(11,20)(12,21)(14,22)(27,34)(30,32)(35,37)(38,40);(1,3,5,4)(2,6,8,7)(9,12)(10,18)(11,13)(14,16)(15,21)(17,24)(19,22)(20,23)(25,28,31,26)(27,30,34,32)(29,33,39,36)(35,40,37,38);(1,3,5,4)(2,6,8,7)(9,13)(10,14)(11,21)(12,20)(15,23)(16,24)(17,22)(18,19)(25,29,31,39)(26,33,28,36)(27,35,34,37)(30,38,32,40);(1,4,5,3)(2,7,8,6)(9,14)(10,13)(11,18)(12,16)(15,22)(17,23)(19,21)(20,24)(25,30)(26,27)(28,34)(29,38)(31,32)(33,35)(36,37)(39,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=hi,c^2=gh,d^2=gi,e^2=gi,f^2=gh,b^a=bg,c^a=ch,e^a=eh,c^b=ci,e^b=ei,d^c=dh,f^c=fi,e^d=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7705298);