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 | |
4511 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,4,7,8)(10,15)(11,18)(17,24)(19,20)(25,26)(27,29)(28,32)(30,34)(31,38)(33,39)(35,40)(36,37);(2,6)(3,7)(9,10)(11,16)(12,17)(13,18)(14,15)(19,21)(20,23)(22,24)(25,27,30,33)(26,29,34,39)(28,36,38,35)(31,40,32,37);(1,3)(2,4)(5,7)(6,8)(25,28,30,38)(26,31,34,32)(27,35,33,36)(29,37,39,40);(1,4)(2,3)(5,8)(6,7)(9,11,14,18)(10,16,15,13)(12,20,22,19)(17,23,24,21)(25,28)(26,32)(27,36)(29,37)(30,38)(31,34)(33,35)(39,40);(9,12,14,22)(10,17,15,24)(11,19,18,20)(13,23,16,21)(25,27)(26,29)(28,35)(30,33)(31,37)(32,40)(34,39)(36,38);(9,13,14,16)(10,18,15,11)(12,21,22,23)(17,19,24,20)(25,29)(26,33)(27,34)(28,37)(30,39)(31,36)(32,35)(38,40)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=h,d^2=ghi,e^2=ghi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10123724);