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 | |
4525 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,7)(9,10)(11,13)(12,15)(14,20)(16,18)(17,22)(19,23)(21,24)(25,26,31,33)(27,32,35,39)(28,30,37,40)(29,34,36,38);(1,2,5,4)(3,7,8,6)(9,11)(10,16)(12,13)(14,19)(15,18)(17,23)(20,21)(22,24)(25,27,31,35)(26,32,33,39)(28,36,37,29)(30,38,40,34);(1,3,5,8)(2,6,4,7)(10,12)(11,18)(17,20)(19,24)(26,33)(27,35)(28,37)(34,38);(1,4,5,2)(3,7,8,6)(9,12,15,10)(11,13,18,16)(14,20,22,17)(19,21,24,23)(25,28)(26,30)(27,36)(29,35)(31,37)(32,38)(33,40)(34,39);(9,13)(10,11)(12,18)(14,21)(15,16)(17,19)(20,24)(22,23)(25,29,31,36)(26,34,33,38)(27,37,35,28)(30,32,40,39);(9,14)(10,17)(11,19)(12,20)(13,21)(15,22)(16,23)(18,24)(25,30)(26,28)(27,38)(29,39)(31,40)(32,36)(33,37)(34,35)Polycyclic presentation:
2^9|a^2=gi,b^2=gh,c^2=hi,d^2=g,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, 10065830);