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 | |
4530 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,7,5)(3,4,8,6)(9,10)(11,13)(12,20)(14,15)(16,18)(17,21)(19,24)(22,23)(25,26,31,34)(27,30,29,32)(28,33,38,40)(35,37,36,39);(2,5)(3,8)(10,15)(12,21)(13,16)(19,22)(27,29)(30,32)(35,36)(37,39);(1,3)(2,6)(4,5)(7,8)(10,15)(11,18)(17,20)(19,22)(25,27,31,29)(26,32,34,30)(28,36,38,35)(33,37,40,39);(1,4,7,6)(2,3,5,8)(9,11,14,18)(10,16,15,13)(12,22,21,19)(17,23,20,24)(25,28)(26,33)(27,35)(29,36)(30,37)(31,38)(32,39)(34,40);(1,5,7,2)(3,4,8,6)(9,12,14,21)(10,17,15,20)(11,19,18,22)(13,23,16,24)(25,29,31,27)(26,30,34,32)(28,35,38,36)(33,39,40,37);(1,6,7,4)(2,8,5,3)(9,13,14,16)(10,18,15,11)(12,23,21,24)(17,22,20,19)(25,30)(26,27)(28,37)(29,34)(31,32)(33,35)(36,40)(38,39)Polycyclic presentation:
2^9|a^2=i,c^2=gh,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8733465);