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 | |
4519 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)(3,8)(4,6)(5,7)(9,10,14,15)(11,19,20,16)(12,17,21,24)(13,22,23,18)(25,26)(27,30)(28,33)(29,34)(31,32)(35,40)(36,38)(37,39);(1,3,5,4)(2,6,7,8)(10,15)(12,21)(16,19)(18,22)(25,27,31,35)(26,30,32,40)(28,36,37,29)(33,38,39,34);(10,15)(11,20)(17,24)(18,22)(26,32)(27,35)(28,37)(34,38);(2,7)(6,8)(9,11)(10,16)(12,22)(13,17)(14,20)(15,19)(18,21)(23,24)(25,28)(26,33)(27,36)(29,35)(30,38)(31,37)(32,39)(34,40);(1,2)(3,6)(4,8)(5,7)(9,12)(10,17)(11,18)(13,19)(14,21)(15,24)(16,23)(20,22)(25,29,31,36)(26,34,32,38)(27,37,35,28)(30,39,40,33);(1,4,5,3)(2,6,7,8)(9,13,14,23)(10,18,15,22)(11,17,20,24)(12,16,21,19)(25,30)(26,35)(27,32)(28,38)(29,33)(31,40)(34,37)(36,39)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10054185);