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 | |
4165 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,14)(12,13)(15,16)(17,23)(18,19)(20,22)(21,24)(25,26,32,31)(27,35,36,30)(28,33,39,38)(29,40,37,34);(2,3)(6,7)(9,11,15,17)(10,14,16,23)(12,18,20,21)(13,19,22,24)(25,27,32,36)(26,30,31,35)(28,34,39,40)(29,33,37,38);(1,2,5,3)(4,6,8,7)(10,16)(11,17)(12,20)(19,24)(25,28,32,39)(26,33,31,38)(27,34,36,40)(29,30,37,35);(9,12)(10,13)(11,18)(14,19)(15,20)(16,22)(17,21)(23,24)(25,29,32,37)(26,34,31,40)(27,33,36,38)(28,30,39,35);(1,3,5,2)(4,6,8,7)(9,13)(10,12)(11,19)(14,18)(15,22)(16,20)(17,24)(21,23)(25,30)(26,27)(28,37)(29,39)(31,36)(32,35)(33,34)(38,40);(1,4)(2,6)(3,7)(5,8)(9,14)(10,17)(11,16)(12,19)(13,21)(15,23)(18,22)(20,24)(25,31,32,26)(27,35,36,30)(28,38,39,33)(29,40,37,34)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=ghi,d^2=g,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,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, 9755873);