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 | |
4528 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10,15,12)(11,16,18,14)(13,17,23,20)(19,22,21,24)(26,30)(28,36)(31,33)(34,40);(2,3)(6,7)(9,11,15,18)(10,16,12,14)(13,21,23,19)(17,24,20,22)(25,26)(27,33)(28,32)(29,30)(31,35)(34,39)(36,37)(38,40);(1,3,5,2)(4,7,8,6)(10,12)(11,18)(17,20)(19,21)(25,26,29,30)(27,31,35,33)(28,32,36,37)(34,38,40,39);(1,4)(2,6)(3,7)(5,8)(9,12,15,10)(11,16,18,14)(13,17,23,20)(19,24,21,22)(25,26)(27,31)(28,32)(29,30)(33,35)(34,38)(36,37)(39,40);(9,13)(10,17)(11,19)(12,20)(14,24)(15,23)(16,22)(18,21)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,40)(37,39);(9,14,15,16)(10,18,12,11)(13,22,23,24)(17,19,20,21)(25,28,29,36)(26,32,30,37)(27,34,35,40)(31,38,33,39)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=g,d^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10154597);