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 | |
4706 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10,15,17)(11,20,21,16)(12,22,19,14)(13,18,23,24)(25,26,31,33)(27,35,37,32)(28,30,39,40)(29,34,36,38);(2,3)(6,7)(9,11)(10,16)(12,13)(14,18)(15,21)(17,20)(19,23)(22,24)(25,27,31,37)(26,32,33,35)(28,38,39,34)(29,30,36,40);(1,3,5,2)(4,7,8,6)(10,17)(11,21)(12,19)(18,24)(26,33)(28,39)(32,35)(34,38);(1,4,5,8)(2,6,3,7)(9,12,15,19)(10,14,17,22)(11,13,21,23)(16,18,20,24)(25,28)(26,30)(27,34)(29,32)(31,39)(33,40)(35,36)(37,38);(9,13)(10,18)(11,19)(12,21)(14,20)(15,23)(16,22)(17,24)(25,29)(26,34)(27,30)(28,32)(31,36)(33,38)(35,39)(37,40);(9,14)(10,19)(11,18)(12,17)(13,20)(15,22)(16,23)(21,24)(25,30)(26,28)(27,36)(29,37)(31,40)(32,38)(33,39)(34,35)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=gh,d^2=i,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,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, 9854225);