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 | |
4486 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)(11,17)(12,16)(13,14)(15,18)(19,24)(20,23)(21,22)(25,26)(27,36)(28,39)(29,30)(31,32)(33,37)(34,38)(35,40);(1,2,5,4)(3,7,8,6)(11,18)(15,17)(19,20)(23,24)(26,32)(27,37)(29,35)(34,39);(1,3,5,8)(2,6,4,7)(9,11,14,18)(10,15,13,17)(12,20,22,19)(16,24,21,23)(25,27)(26,33)(28,35)(29,38)(30,39)(31,37)(32,36)(34,40);(3,8)(6,7)(9,12,14,22)(10,16,13,21)(11,19,18,20)(15,23,17,24)(25,28,31,38)(26,34,32,39)(27,35,37,29)(30,33,40,36);(1,3)(2,7)(4,6)(5,8)(9,11)(10,17)(12,20)(13,15)(14,18)(16,23)(19,22)(21,24)(25,29,31,35)(26,30,32,40)(27,38,37,28)(33,39,36,34);(1,4,5,2)(3,6,8,7)(9,13,14,10)(11,15,18,17)(12,21,22,16)(19,23,20,24)(25,30)(26,35)(27,39)(28,33)(29,32)(31,40)(34,37)(36,38)Polycyclic presentation:
2^9|b^2=hi,c^2=gi,d^2=h,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8935312);