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 | |
4334 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)(11,14)(12,18)(13,16)(15,19)(17,23)(20,24)(21,22)(28,36)(31,37)(34,40)(38,39);(2,3)(6,7)(9,11,15,13)(10,16,19,14)(12,20,23,21)(17,24,18,22)(25,26)(27,32)(28,37)(29,33)(30,35)(31,36)(34,38)(39,40);(1,3,5,2)(4,7,8,6)(9,12)(10,17)(11,20)(13,21)(14,22)(15,23)(16,24)(18,19)(25,27)(26,30)(28,34)(29,35)(31,38)(32,33)(36,40)(37,39);(1,4,5,8)(2,6,3,7)(9,12)(10,18)(11,21)(13,20)(14,22)(15,23)(16,24)(17,19)(25,28)(26,31)(27,34)(29,36)(30,38)(32,39)(33,37)(35,40);(9,13,15,11)(10,16,19,14)(12,21,23,20)(17,24,18,22)(25,27)(26,32)(28,34)(29,35)(30,33)(31,39)(36,40)(37,38);(9,14,15,16)(10,13,19,11)(12,22,23,24)(17,21,18,20)(25,26)(27,30)(28,31)(29,33)(32,35)(34,38)(36,37)(39,40)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=gh,d^2=gh,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10362867);