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 | |
4462 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)(3,7)(4,8)(5,6)(9,10,13,14)(11,17,18,15)(12,16,21,22)(19,24,20,23)(25,26,31,33)(27,32,37,40)(28,34,35,29)(30,36,39,38);(2,6)(3,8)(10,14)(11,18)(16,22)(19,20)(25,27,31,37)(26,32,33,40)(28,36,35,38)(29,30,34,39);(1,3)(2,4)(5,8)(6,7)(10,14)(15,17)(16,22)(23,24)(26,33)(27,37)(29,34)(36,38);(1,3)(2,4)(5,8)(6,7)(9,11)(10,15)(12,20)(13,18)(14,17)(16,24)(19,21)(22,23)(25,28,31,35)(26,34,33,29)(27,38,37,36)(30,40,39,32);(1,4)(2,3)(5,7)(6,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,23)(20,24)(21,22)(25,29,31,34)(26,35,33,28)(27,39,37,30)(32,36,40,38);(9,12,13,21)(10,16,14,22)(11,19,18,20)(15,23,17,24)(25,30,31,39)(26,36,33,38)(27,34,37,29)(28,32,35,40)Polycyclic presentation:
2^9|a^2=gi,b^2=gh,d^2=gh,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8542009);