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 | |
4323 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,17)(11,18,14,16)(12,21,23,13)(19,20,22,24)(25,26)(27,33)(28,36)(29,30)(31,35)(32,37)(34,38)(39,40);(1,2)(3,8)(4,6)(5,7)(9,11,15,14)(10,16,17,18)(12,19,23,22)(13,20,21,24)(26,30)(31,33)(32,36)(38,39);(1,3,5,6)(2,4,7,8)(25,27,29,35)(26,31,30,33)(28,34,37,40)(32,38,36,39);(9,12)(10,13)(11,19)(14,22)(15,23)(16,20)(17,21)(18,24)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,39)(37,40);(1,2)(3,4)(5,7)(6,8)(9,13)(10,12)(11,20)(14,24)(15,21)(16,19)(17,23)(18,22)(25,26)(27,31)(28,36)(29,30)(32,37)(33,35)(34,39)(38,40);(1,4)(2,6)(3,7)(5,8)(9,14,15,11)(10,16,17,18)(12,22,23,19)(13,20,21,24)(25,28,29,37)(26,32,30,36)(27,34,35,40)(31,38,33,39)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=hi,f^2=g,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, 9588905);