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 | |
4398 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,4)(5,8)(6,7)(9,10,15,19)(11,20,22,16)(12,14,18,23)(13,17,21,24)(25,26,31,35)(27,34,30,32)(28,33,39,29)(36,40,38,37);(1,3,5,7)(2,4,8,6)(9,11)(10,16)(12,21)(13,18)(14,17)(15,22)(19,20)(23,24)(25,27,31,30)(26,32,35,34)(28,36,39,38)(29,40,33,37);(1,3)(2,6)(4,8)(5,7)(25,28,31,39)(26,29,35,33)(27,36,30,38)(32,40,34,37);(1,2,5,8)(3,4,7,6)(9,12,15,18)(10,14,19,23)(11,21,22,13)(16,17,20,24)(25,28)(26,33)(27,36)(29,35)(30,38)(31,39)(32,37)(34,40);(1,3)(2,4)(5,7)(6,8)(9,13,15,21)(10,17,19,24)(11,18,22,12)(14,20,23,16)(25,29,31,33)(26,28,35,39)(27,37,30,40)(32,38,34,36);(1,4)(2,7)(3,8)(5,6)(9,14)(10,18)(11,17)(12,19)(13,16)(15,23)(20,21)(22,24)(25,30,31,27)(26,34,35,32)(28,38,39,36)(29,37,33,40)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=hi,d^2=gh,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,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, 9633618);