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 | |
4660 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,6,8,4)(9,10)(11,18)(12,20)(13,15)(14,17)(16,21)(19,23)(22,24)(25,26)(27,30)(28,38)(29,40)(31,35)(32,37)(33,39)(34,36);(1,3)(2,6)(4,7)(5,8)(9,11,14,13)(10,15,17,18)(12,19,21,22)(16,23,20,24)(25,27)(26,32)(28,29)(30,35)(31,37)(33,40)(34,38)(36,39);(1,2)(3,4)(5,7)(6,8)(12,21)(16,20)(19,22)(23,24)(25,28,31,39)(26,33,35,38)(27,29,37,36)(30,34,32,40);(1,3,5,8)(2,6,7,4)(9,12)(10,16)(11,19)(13,22)(14,21)(15,23)(17,20)(18,24)(25,28)(26,33)(27,36)(29,37)(30,40)(31,39)(32,34)(35,38);(1,2)(3,6)(4,8)(5,7)(9,13,14,11)(10,15,17,18)(12,22,21,19)(16,23,20,24)(25,29,31,36)(26,34,35,40)(27,28,37,39)(30,33,32,38);(1,4,5,6)(2,3,7,8)(9,10)(11,15)(12,20)(13,18)(14,17)(16,21)(19,24)(22,23)(25,30,31,32)(26,27,35,37)(28,34,39,40)(29,38,36,33)Polycyclic presentation:
2^9|a^2=i,b^2=ghi,c^2=hi,d^2=i,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8627983);