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 | |
4577 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,16)(11,18,13,15)(12,17,22,21)(19,24,20,23)(25,26,31,28)(27,32,35,36)(29,33,39,38)(30,37,40,34);(1,2)(3,4)(5,7)(6,8)(9,11,14,13)(10,15,16,18)(12,20,22,19)(17,24,21,23)(25,27)(26,32)(28,36)(29,34)(30,33)(31,35)(37,39)(38,40);(1,3,5,8)(2,6,7,4)(10,16)(15,18)(17,21)(23,24)(26,28)(27,35)(33,38)(34,37);(3,8)(4,6)(9,10)(11,15)(12,21)(13,18)(14,16)(17,22)(19,24)(20,23)(25,28,31,26)(27,36,35,32)(29,33,39,38)(30,37,40,34);(1,4)(2,3)(5,6)(7,8)(9,12,14,22)(10,17,16,21)(11,19,13,20)(15,23,18,24)(25,29)(26,33)(27,37)(28,38)(30,36)(31,39)(32,40)(34,35);(1,2)(3,6)(4,8)(5,7)(9,13,14,11)(10,18,16,15)(12,19,22,20)(17,23,21,24)(25,30)(26,34)(27,33)(28,37)(29,32)(31,40)(35,38)(36,39)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=ghi,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9044675);