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 | |
4359 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,7)(3,5,4,8)(9,10,15,14)(11,18,13,16)(12,17,20,22)(19,24,21,23)(25,26)(27,35)(28,39)(29,33)(30,34)(31,38)(32,37)(36,40);(2,7)(3,4)(9,11,15,13)(10,16,14,18)(12,19,20,21)(17,23,22,24)(25,27)(26,31)(28,36)(29,37)(30,38)(32,40)(33,39)(34,35);(1,3,6,4)(2,8,7,5)(12,20)(17,22)(19,21)(23,24)(25,26,30,34)(27,31,38,35)(28,32,37,39)(29,33,36,40);(1,4,6,3)(2,5,7,8)(9,12)(10,17)(11,19)(13,21)(14,22)(15,20)(16,23)(18,24)(25,26)(27,35)(28,32)(29,40)(30,34)(31,38)(33,36)(37,39);(1,5)(2,3)(4,7)(6,8)(9,13,15,11)(10,16,14,18)(12,21,20,19)(17,23,22,24)(25,28)(26,32)(27,36)(29,38)(30,37)(31,40)(33,35)(34,39);(9,14,15,10)(11,18,13,16)(12,17,20,22)(19,23,21,24)(25,29,30,36)(26,33,34,40)(27,37,38,28)(31,39,35,32)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=h,d^2=i,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8658020);