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 | |
4551 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,15)(12,20)(13,14)(16,21)(17,18)(19,23)(22,24)(25,26,30,34)(27,28,32,31)(29,33,38,40)(35,36,37,39);(1,2,4,6)(3,5,8,7)(9,10)(11,17)(12,20)(13,14)(15,18)(16,21)(19,22)(23,24)(25,27,30,32)(26,31,34,28)(29,37,38,35)(33,36,40,39);(1,2)(3,5)(4,6)(7,8)(10,14)(11,18)(16,20)(19,24)(25,26,30,34)(27,31,32,28)(29,33,38,40)(35,39,37,36);(1,3,4,8)(2,5,6,7)(9,10)(11,15)(12,16)(13,14)(17,18)(19,22)(20,21)(23,24)(25,26)(27,31)(28,32)(29,33)(30,34)(35,39)(36,37)(38,40);(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,23)(21,24)(25,28,30,31)(26,32,34,27)(29,36,38,39)(33,37,40,35);(9,12,13,21)(10,16,14,20)(11,19,18,24)(15,22,17,23)(25,29)(26,33)(27,35)(28,36)(30,38)(31,39)(32,37)(34,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=g,c^2=ghi,d^2=hi,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,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, 8714052);