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 | |
4556 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)(3,7)(4,6)(5,8)(9,10,14,15)(11,17,18,12)(13,21,22,16)(19,20,24,23)(26,31)(27,33)(32,39)(34,35);(1,2,4,6)(3,5,8,7)(10,15)(12,17)(16,21)(20,23)(25,26)(27,29)(28,32)(30,31)(33,38)(34,36)(35,40)(37,39);(1,2)(3,5)(4,6)(7,8)(10,15)(11,18)(13,22)(20,23)(27,33)(29,38)(34,35)(36,40);(1,3)(2,5)(4,8)(6,7)(9,11,14,18)(10,12,15,17)(13,19,22,24)(16,20,21,23)(25,27)(26,29)(28,35)(30,33)(31,38)(32,40)(34,37)(36,39);(9,12)(10,11)(13,20)(14,17)(15,18)(16,19)(21,24)(22,23)(25,28,30,37)(26,32,31,39)(27,34,33,35)(29,36,38,40);(9,13,14,22)(10,16,15,21)(11,19,18,24)(12,20,17,23)(25,29)(26,27)(28,36)(30,38)(31,33)(32,34)(35,39)(37,40)Polycyclic presentation:
2^9|a^2=h,b^2=i,d^2=h,e^2=ghi,f^2=h,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, 8826699);