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 | |
4411 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,14)(11,12,16,19)(13,21,22,17)(18,24,23,20)(25,26,30,34)(27,32,28,31)(29,33,37,40)(35,39,36,38);(1,2,5,4)(3,7,8,6)(9,11)(10,12)(13,18)(14,19)(15,16)(17,20)(21,24)(22,23)(25,26)(27,31)(28,32)(29,33)(30,34)(35,38)(36,39)(37,40);(2,4)(6,7)(10,14)(11,16)(17,21)(18,23)(25,27,30,28)(26,31,34,32)(29,36,37,35)(33,39,40,38);(1,3)(2,6)(4,7)(5,8)(9,12,15,19)(10,16,14,11)(13,20,22,24)(17,23,21,18)(25,26)(27,31)(28,32)(29,33)(30,34)(35,38)(36,39)(37,40);(9,13,15,22)(10,17,14,21)(11,18,16,23)(12,20,19,24)(25,28,30,27)(26,32,34,31)(29,36,37,35)(33,39,40,38);(1,4,5,2)(3,7,8,6)(9,14,15,10)(11,19,16,12)(13,21,22,17)(18,24,23,20)(25,29,30,37)(26,33,34,40)(27,35,28,36)(31,38,32,39)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,c^2=g,d^2=gh,e^2=h,f^2=gi,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, 8103003);