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 | |
4575 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)(3,4)(5,8)(6,7)(9,10)(11,14)(12,19)(13,15)(16,20)(17,23)(18,21)(22,24)(25,26,30,35)(27,32,38,31)(28,33,39,37)(29,36,40,34);(1,3)(2,4)(5,6)(7,8)(9,11,13,18)(10,14,15,21)(12,17,20,24)(16,22,19,23)(25,27,30,38)(26,31,35,32)(28,36,39,34)(29,37,40,33);(1,4)(2,6)(3,8)(5,7)(10,15)(14,21)(16,19)(22,23)(25,27,30,38)(26,31,35,32)(28,36,39,34)(29,37,40,33);(1,2,5,8)(3,7,6,4)(9,10)(11,14)(12,16)(13,15)(17,22)(18,21)(19,20)(23,24)(25,27)(26,32)(28,36)(29,37)(30,38)(31,35)(33,40)(34,39);(1,4)(2,3)(5,7)(6,8)(9,11)(10,14)(12,17)(13,18)(15,21)(16,22)(19,23)(20,24)(25,28)(26,33)(27,36)(29,31)(30,39)(32,40)(34,38)(35,37);(1,3)(2,7)(4,8)(5,6)(9,12)(10,16)(11,17)(13,20)(14,22)(15,19)(18,24)(21,23)(25,29,30,40)(26,34,35,36)(27,37,38,33)(28,32,39,31)Polycyclic presentation:
2^9|a^2=g,b^2=hi,c^2=g,d^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9854172);