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 | |
4594 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,8,4,6)(9,10)(11,19)(12,21)(13,18)(14,22)(15,17)(16,20)(23,24)(25,26,31,33)(27,28,36,32)(29,34,40,38)(30,37,39,35);(3,4)(6,8)(9,11,15,20)(10,16,17,19)(12,13,22,23)(14,18,21,24)(25,27,31,36)(26,32,33,28)(29,39,40,30)(34,37,38,35);(1,3,5,4)(2,6,7,8)(10,17)(11,20)(12,22)(18,24)(26,33)(28,32)(34,38)(35,37);(3,4)(6,8)(9,12)(10,14)(11,13)(15,22)(16,18)(17,21)(19,24)(20,23)(25,28)(26,27)(29,35)(30,38)(31,32)(33,36)(34,39)(37,40);(1,4,5,3)(2,6,7,8)(9,13)(10,18)(11,12)(14,16)(15,23)(17,24)(19,21)(20,22)(25,29,31,40)(26,34,33,38)(27,30,36,39)(28,37,32,35);(1,2)(3,6)(4,8)(5,7)(9,14)(10,12)(11,18)(13,16)(15,21)(17,22)(19,23)(20,24)(25,30)(26,35)(27,29)(28,38)(31,39)(32,34)(33,37)(36,40)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=gh,e^2=gi,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, 10054220);