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 | |
4465 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(3,7)(9,10,14,17)(11,15,18,12)(13,16,22,24)(19,23,21,20)(25,26,30,31)(27,32,35,29)(28,36,39,33)(34,38,37,40);(2,5)(3,7)(9,11,14,18)(10,15,17,12)(13,21,22,19)(16,20,24,23)(26,31)(29,32)(33,36)(38,40);(1,2)(3,6)(4,5)(7,8)(11,18)(12,15)(19,21)(20,23)(25,27,30,35)(26,32,31,29)(28,34,39,37)(33,40,36,38);(2,5)(6,8)(9,12)(10,11)(13,20)(14,15)(16,19)(17,18)(21,24)(22,23)(25,27)(26,29)(28,37)(30,35)(31,32)(33,40)(34,39)(36,38);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,18)(13,16)(14,17)(19,23)(20,21)(22,24)(25,28)(26,33)(27,34)(29,38)(30,39)(31,36)(32,40)(35,37);(1,3,4,7)(2,6,5,8)(9,13)(10,16)(11,19)(12,20)(14,22)(15,23)(17,24)(18,21)(25,29)(26,27)(28,38)(30,32)(31,35)(33,34)(36,37)(39,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=hi,c^2=g,f^2=h,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, 8735864);