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 | |
4171 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10)(11,16)(12,20)(13,17)(14,19)(15,21)(18,22)(23,24)(25,26,31,33)(27,37,38,32)(28,29,35,34)(30,40,39,36);(2,3)(6,7)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24)(25,27,31,38)(26,32,33,37)(28,36,35,40)(29,39,34,30);(1,3,5,2)(4,7,8,6)(26,33)(28,35)(32,37)(36,40);(1,4,5,8)(2,6,3,7)(9,12,13,21)(10,15,17,20)(11,18,19,24)(14,22,16,23)(25,28,31,35)(26,34,33,29)(27,36,38,40)(30,37,39,32);(9,11)(10,16)(12,18)(13,19)(14,17)(15,23)(20,22)(21,24)(25,29,31,34)(26,35,33,28)(27,30,38,39)(32,36,37,40);(9,10)(11,14)(12,15)(13,17)(16,19)(18,22)(20,21)(23,24)(25,30)(26,36)(27,29)(28,37)(31,39)(32,35)(33,40)(34,38)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=i,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9592740);