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 | |
4365 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,4)(5,7)(6,8)(9,10,14,15)(11,16,18,12)(13,21,23,17)(19,20,24,22)(25,26)(27,31)(28,34)(29,37)(30,32)(33,36)(35,40)(38,39);(1,3)(2,6)(4,7)(5,8)(10,15)(12,16)(17,21)(20,22)(25,27)(26,31)(28,35)(29,38)(30,36)(32,33)(34,40)(37,39);(10,15)(11,18)(17,21)(19,24)(26,32)(27,36)(34,38)(35,37);(9,11)(10,16)(12,15)(13,19)(14,18)(17,22)(20,21)(23,24)(25,27)(26,33)(28,37)(29,38)(30,36)(31,32)(34,40)(35,39);(1,4,5,6)(2,3,7,8)(9,12,14,16)(10,11,15,18)(13,22,23,20)(17,24,21,19)(25,28)(26,34)(27,37)(29,31)(30,39)(32,38)(33,40)(35,36);(1,3)(2,4)(5,8)(6,7)(9,13,14,23)(10,17,15,21)(11,19,18,24)(12,20,16,22)(25,29,30,40)(26,35,32,37)(27,38,36,34)(28,33,39,31)Polycyclic presentation:
2^9|a^2=hi,e^2=g,f^2=h,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, 9625728);