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 | |
4301 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,19)(12,17)(13,21)(14,15)(16,20)(18,23)(22,24)(25,26,30,33)(27,31,36,39)(28,29,37,32)(34,35,40,38);(1,2)(3,8)(4,6)(5,7)(10,15)(12,22)(16,19)(18,21)(25,27,30,36)(26,31,33,39)(28,34,37,40)(29,35,32,38);(1,3,5,6)(2,4,7,8)(10,15)(11,20)(17,24)(18,21)(25,28,30,37)(26,32,33,29)(27,34,36,40)(31,38,39,35);(9,11,14,20)(10,16,15,19)(12,21,22,18)(13,17,23,24)(25,28)(26,29)(27,34)(30,37)(31,35)(32,33)(36,40)(38,39);(1,2)(3,4)(5,7)(6,8)(9,12,14,22)(10,17,15,24)(11,21,20,18)(13,16,23,19)(25,27)(26,31)(28,34)(29,35)(30,36)(32,38)(33,39)(37,40);(1,4)(2,6)(3,7)(5,8)(9,13,14,23)(10,18,15,21)(11,17,20,24)(12,19,22,16)(25,29)(26,28)(27,35)(30,32)(31,34)(33,37)(36,38)(39,40)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=ghi,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,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, 9588932);