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 | |
4528 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,13)(11,12,19,16)(14,17,22,21)(18,20,24,23)(25,26,31,34)(27,32,36,40)(28,37,35,33)(29,39,38,30);(1,2)(3,7)(4,5)(6,8)(9,11,15,19)(10,16,13,12)(14,18,22,24)(17,23,21,20)(25,27,31,36)(26,32,34,40)(28,38,35,29)(30,33,39,37);(2,5)(6,7)(14,22)(17,21)(18,24)(20,23)(25,28,31,35)(26,33,34,37)(27,29,36,38)(30,40,39,32);(3,8)(6,7)(9,12)(10,11)(13,19)(14,20)(15,16)(17,18)(21,24)(22,23)(25,27)(26,32)(28,29)(30,33)(31,36)(34,40)(35,38)(37,39);(1,3)(2,6)(4,8)(5,7)(9,13,15,10)(11,16,19,12)(14,21,22,17)(18,23,24,20)(25,29)(26,30)(27,35)(28,36)(31,38)(32,37)(33,40)(34,39);(1,2)(3,6)(4,5)(7,8)(9,14,15,22)(10,17,13,21)(11,18,19,24)(12,20,16,23)(25,30)(26,29)(27,33)(28,32)(31,39)(34,38)(35,40)(36,37)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=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, 8254084);