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 | |
4315 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,7)(5,8)(9,10)(11,13)(12,20)(14,15)(16,18)(17,22)(19,23)(21,24)(25,26,30,34)(27,28,32,31)(29,37,38,33)(35,39,40,36);(1,2)(3,5)(4,6)(7,8)(10,15)(11,18)(17,20)(19,24)(25,27,30,32)(26,31,34,28)(29,35,38,40)(33,39,37,36);(1,2,4,6)(3,5,7,8)(9,11)(10,16)(12,19)(13,15)(14,18)(17,23)(20,21)(22,24)(25,26,30,34)(27,31,32,28)(29,33,38,37)(35,39,40,36);(1,2)(3,5)(4,6)(7,8)(9,11)(10,16)(12,19)(13,15)(14,18)(17,23)(20,21)(22,24)(25,26)(27,31)(28,32)(29,33)(30,34)(35,39)(36,40)(37,38);(1,2)(3,5)(4,6)(7,8)(9,12,14,22)(10,17,15,20)(11,19,18,24)(13,21,16,23)(25,28,30,31)(26,32,34,27)(29,36,38,39)(33,40,37,35);(1,3)(2,5)(4,7)(6,8)(9,13,14,16)(10,11,15,18)(12,21,22,23)(17,19,20,24)(25,29)(26,33)(27,35)(28,36)(30,38)(31,39)(32,40)(34,37)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=gi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8034321);