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 | |
4348 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,4)(3,7,5,8)(9,10,14,15)(11,13,18,16)(12,20,23,17)(19,24,21,22)(25,26)(27,29)(28,33)(30,35)(31,32)(34,39)(36,38)(37,40);(1,3,6,5)(2,7,4,8)(10,15)(11,18)(17,20)(19,21)(25,27)(26,29)(28,30)(31,34)(32,39)(33,35)(36,37)(38,40);(1,3,6,5)(2,8,4,7)(10,15)(13,16)(17,20)(22,24)(26,32)(27,34)(30,36)(33,40);(1,4,6,2)(3,7,5,8)(9,11,14,18)(10,16,15,13)(12,21,23,19)(17,22,20,24)(25,28)(26,33)(27,36)(29,38)(30,34)(31,37)(32,40)(35,39);(1,5,6,3)(2,7,4,8)(9,12,14,23)(10,17,15,20)(11,19,18,21)(13,22,16,24)(25,29,31,39)(26,34,32,27)(28,35,37,38)(30,33,36,40);(9,13,14,16)(10,18,15,11)(12,22,23,24)(17,21,20,19)(25,30,31,36)(26,35,32,38)(27,28,34,37)(29,33,39,40)Polycyclic presentation:
2^9|a^2=hi,b^2=ghi,c^2=ghi,d^2=hi,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8247101);