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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4578 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,14)(12,15)(13,16)(17,20)(18,21)(19,22)(23,24)(25,26)(27,30)(28,32)(29,37)(31,34)(33,39)(35,40)(36,38);(1,2,5,6)(3,7,8,4)(10,11)(15,17)(16,18)(22,23)(25,26)(27,30)(28,32)(29,33)(31,34)(35,38)(36,40)(37,39);(2,6)(4,7)(13,21)(16,18)(19,24)(22,23)(25,27,31,28)(26,32,34,30)(29,35,39,36)(33,40,37,38);(1,3)(2,4)(5,8)(6,7)(9,11,14,10)(12,17,20,15)(13,18,21,16)(19,23,24,22)(25,28,31,27)(26,32,34,30)(29,36,39,35)(33,40,37,38);(9,12,14,20)(10,15,11,17)(13,19,21,24)(16,22,18,23)(25,29,31,39)(26,33,34,37)(27,35,28,36)(30,38,32,40);(1,4)(2,3)(5,7)(6,8)(9,13)(10,16)(11,18)(12,19)(14,21)(15,22)(17,23)(20,24)(25,30,31,32)(26,27,34,28)(29,38,39,40)(33,35,37,36)Polycyclic presentation:
2^9|b^2=hi,c^2=h,d^2=gh,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8090212);