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 | |
4422 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,7,8,4)(9,10)(11,19)(12,17)(13,22)(14,16)(15,18)(20,24)(21,23)(25,26)(27,34)(28,33)(29,32)(30,31)(35,40)(36,39)(37,38);(1,3)(2,4)(5,8)(6,7)(9,11,15,14)(10,16,18,19)(12,20,13,21)(17,23,22,24)(27,29)(32,34)(35,36)(39,40);(1,2)(3,4)(5,6)(7,8)(9,12,15,13)(10,17,18,22)(11,20,14,21)(16,23,19,24)(25,27,31,29)(26,32,30,34)(28,36,38,35)(33,40,37,39);(2,6)(4,7)(9,13,15,12)(10,17,18,22)(11,21,14,20)(16,23,19,24)(25,28,31,38)(26,33,30,37)(27,35,29,36)(32,39,34,40);(1,4,5,7)(2,3,6,8)(9,10)(11,16)(12,17)(13,22)(14,19)(15,18)(20,23)(21,24)(25,29,31,27)(26,32,30,34)(28,35,38,36)(33,40,37,39);(1,3)(2,4)(5,8)(6,7)(9,14,15,11)(10,16,18,19)(12,21,13,20)(17,23,22,24)(25,30,31,26)(27,34,29,32)(28,37,38,33)(35,40,36,39)Polycyclic presentation:
2^9|a^2=hi,b^2=ghi,c^2=gh,d^2=gh,e^2=h,f^2=gh,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, 10009444);