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 | |
4394 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,17)(12,16)(13,18)(14,15)(19,24)(20,21)(22,23)(25,26,31,35)(27,36,37,32)(28,38,39,33)(29,30,34,40);(1,2)(3,8)(4,6)(5,7)(9,11)(10,13)(12,20)(14,18)(15,17)(16,24)(19,22)(21,23)(25,27,31,37)(26,32,35,36)(28,34,39,29)(30,33,40,38);(1,3,5,6)(2,4,7,8)(10,15)(11,18)(16,23)(19,20);(1,2,5,7)(3,4,6,8)(9,11)(10,13)(12,19)(14,18)(15,17)(16,21)(20,22)(23,24)(25,28,31,39)(26,33,35,38)(27,29,37,34)(30,36,40,32);(1,2)(3,4)(5,7)(6,8)(9,12)(10,16)(11,19)(13,21)(14,22)(15,23)(17,24)(18,20)(25,29)(26,30)(27,28)(31,34)(32,33)(35,40)(36,38)(37,39);(1,4)(2,6)(3,7)(5,8)(9,13)(10,11)(12,21)(14,17)(15,18)(16,19)(20,23)(22,24)(25,30)(26,34)(27,33)(28,36)(29,35)(31,40)(32,39)(37,38)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=hi,d^2=gi,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, 10157542);