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 | |
4467 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,18)(12,20)(13,15)(14,16)(17,22)(19,23)(21,24)(25,26,31,30)(27,28,32,35)(29,37,38,33)(34,40,39,36);(1,2,5,6)(3,7,8,4)(9,11,14,13)(10,15,16,18)(12,19,22,21)(17,23,20,24)(25,27)(26,28)(29,34)(30,35)(31,32)(33,36)(37,40)(38,39);(2,6)(4,7)(10,16)(15,18)(17,20)(23,24)(26,30)(27,32)(33,37)(34,39);(1,3)(2,4)(5,8)(6,7)(9,12)(10,17)(11,19)(13,21)(14,22)(15,23)(16,20)(18,24)(25,28,31,35)(26,32,30,27)(29,36,38,40)(33,39,37,34);(9,13,14,11)(10,18,16,15)(12,21,22,19)(17,24,20,23)(25,29,31,38)(26,33,30,37)(27,34,32,39)(28,36,35,40);(1,4)(2,3)(5,7)(6,8)(9,10)(11,15)(12,17)(13,18)(14,16)(19,23)(20,22)(21,24)(25,30,31,26)(27,35,32,28)(29,37,38,33)(34,40,39,36)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,d^2=gh,e^2=h,f^2=gh,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, 8090101);