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 | |
4909 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,5)(6,8)(9,10,14,15)(11,13,17,21)(12,16,19,22)(18,20,23,24)(25,26)(27,30)(28,36)(29,32)(31,38)(33,35)(34,40)(37,39);(2,5)(6,7)(12,19)(16,22)(18,23)(20,24)(25,26)(27,33)(28,31)(29,32)(30,35)(34,40)(36,38)(37,39);(1,2)(3,6)(4,5)(7,8)(10,15)(11,17)(16,22)(18,23)(25,26,29,32)(27,30,35,33)(28,31,38,36)(34,39,37,40);(1,3)(2,6)(4,8)(5,7)(9,11)(10,13)(12,18)(14,17)(15,21)(16,20)(19,23)(22,24)(25,26)(27,33)(28,31)(29,32)(30,35)(34,40)(36,38)(37,39);(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,27,29,35)(26,30,32,33)(28,37,38,34)(31,40,36,39);(9,13)(10,11)(12,20)(14,21)(15,17)(16,18)(19,24)(22,23)(25,28)(26,31)(27,34)(29,38)(30,39)(32,36)(33,40)(35,37)Polycyclic presentation:
2^9|a^2=ghi,c^2=gi,e^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10065610);