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 | |
4406 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)(11,12)(13,21)(14,15)(16,18)(17,22)(19,23)(20,24)(25,26)(27,32)(28,29)(30,38)(31,33)(34,39)(35,37)(36,40);(2,5)(3,8)(10,15)(12,16)(17,21)(20,23)(25,27,31,28)(26,32,33,29)(30,35,39,36)(34,40,38,37);(1,3)(2,6)(4,5)(7,8)(10,15)(11,18)(13,22)(20,23)(26,33)(27,28)(30,39)(37,40);(1,4,7,6)(2,3,5,8)(9,11,14,18)(10,16,15,12)(13,19,22,24)(17,23,21,20)(25,28,31,27)(26,29,33,32)(30,36,39,35)(34,37,38,40);(1,5,7,2)(3,4,8,6)(9,12,14,16)(10,11,15,18)(13,20,22,23)(17,19,21,24)(25,29)(26,28)(27,33)(30,37)(31,32)(34,36)(35,38)(39,40);(1,6,7,4)(2,8,5,3)(9,13,14,22)(10,17,15,21)(11,19,18,24)(12,20,16,23)(25,30)(26,34)(27,35)(28,36)(29,37)(31,39)(32,40)(33,38)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,d^2=h,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,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, 8812371);