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 | |
4332 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,8)(25,26,31,36)(27,37,40,32)(28,39,38,33)(29,34,35,30);(3,7)(6,8)(9,10,14,11)(12,15,20,17)(13,16,21,18)(19,22,24,23)(25,27,31,40)(26,32,36,37)(28,35,38,29)(30,39,34,33);(1,2,5,4)(3,6,7,8)(10,11)(13,21)(15,17)(19,24)(25,28,31,38)(26,33,36,39)(27,35,40,29)(30,37,34,32);(9,11,14,10)(12,15,20,17)(13,18,21,16)(19,22,24,23)(25,28)(26,33)(27,35)(29,40)(30,32)(31,38)(34,37)(36,39);(1,3,5,7)(2,6,4,8)(9,12,14,20)(10,15,11,17)(13,19,21,24)(16,22,18,23)(25,29,31,35)(26,34,36,30)(27,38,40,28)(32,39,37,33);(1,4,5,2)(3,8,7,6)(9,13)(10,16)(11,18)(12,19)(14,21)(15,22)(17,23)(20,24)(25,30,31,34)(26,35,36,29)(27,39,40,33)(28,32,38,37)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=h,d^2=i,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8830866);