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 | |
4537 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,12)(11,15,19,17)(13,20,21,16)(18,23,24,22)(25,26,31,32)(27,37,39,33)(28,38,40,34)(29,35,36,30);(1,2,5,6)(3,7,8,4)(9,11,14,19)(10,15,12,17)(13,18,21,24)(16,22,20,23)(26,32)(28,40)(29,36)(33,37);(1,3)(2,4)(5,8)(6,7)(10,12)(15,17)(16,20)(22,23)(25,27,31,39)(26,33,32,37)(28,36,40,29)(30,38,35,34);(2,6)(4,7)(9,12,14,10)(11,17,19,15)(13,20,21,16)(18,23,24,22)(25,28,31,40)(26,34,32,38)(27,29,39,36)(30,33,35,37);(1,4)(2,3)(5,7)(6,8)(9,11)(10,15)(12,17)(13,18)(14,19)(16,22)(20,23)(21,24)(25,29)(26,35)(27,28)(30,32)(31,36)(33,34)(37,38)(39,40);(1,3)(2,7)(4,6)(5,8)(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,30,31,35)(26,36,32,29)(27,38,39,34)(28,33,40,37)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,c^2=g,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10133236);