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 | |
4546 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,7,8,4)(9,10)(11,16)(12,18)(13,19)(14,24)(15,17)(20,23)(21,22)(26,31)(27,35)(28,38)(34,39);(2,6)(4,7)(9,11,15,14)(10,16,17,24)(12,20,22,19)(13,18,23,21)(25,26)(27,32)(28,33)(29,39)(30,31)(34,36)(35,37)(38,40);(1,2)(3,4)(5,6)(7,8)(10,17)(11,14)(12,22)(13,23)(25,27,30,35)(26,32,31,37)(28,29,38,36)(33,39,40,34);(1,3)(2,4)(5,8)(6,7)(9,12)(10,18)(11,20)(13,24)(14,19)(15,22)(16,23)(17,21)(25,28,30,38)(26,33,31,40)(27,36,35,29)(32,34,37,39);(1,4)(2,3)(5,7)(6,8)(9,13,15,23)(10,19,17,20)(11,21,14,18)(12,16,22,24)(25,29,30,36)(26,34,31,39)(27,28,35,38)(32,40,37,33);(1,3)(2,4)(5,8)(6,7)(9,14,15,11)(10,16,17,24)(12,19,22,20)(13,18,23,21)(25,26)(27,37)(28,33)(29,34)(30,31)(32,35)(36,39)(38,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=hi,d^2=hi,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8922835);