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 | |
4437 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,18)(12,15)(13,20)(14,16)(17,22)(19,23)(21,24)(25,26)(27,32)(28,34)(29,33)(30,35)(31,36)(37,38)(39,40);(1,3)(2,6)(4,7)(5,8)(9,11,15,14)(10,16,12,18)(13,19,22,21)(17,23,20,24)(25,27,31,29)(26,32,36,33)(28,39,35,37)(30,38,34,40);(9,10,15,12)(11,18,14,16)(13,17,22,20)(19,24,21,23)(25,27,31,29)(26,33,36,32)(28,37,35,39)(30,38,34,40);(1,2,5,4)(3,6,8,7)(9,12,15,10)(11,16,14,18)(13,20,22,17)(19,23,21,24)(25,28,31,35)(26,34,36,30)(27,37,29,39)(32,38,33,40);(1,4,5,2)(3,7,8,6)(9,13)(10,17)(11,19)(12,20)(14,21)(15,22)(16,23)(18,24)(25,29,31,27)(26,32,36,33)(28,39,35,37)(30,40,34,38);(1,3)(2,7)(4,6)(5,8)(9,14,15,11)(10,16,12,18)(13,21,22,19)(17,23,20,24)(25,30,31,34)(26,35,36,28)(27,38,29,40)(32,37,33,39)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=h,d^2=ghi,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8594541);