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 | |
4349 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,18)(11,12,16,15)(13,17,22,24)(19,20,21,23)(25,26,31,28)(27,34,33,30)(29,32,38,36)(35,40,39,37);(1,2)(3,8)(4,5)(6,7)(9,11,14,16)(10,15,18,12)(13,21,22,19)(17,20,24,23)(25,26,31,28)(27,30,33,34)(29,36,38,32)(35,40,39,37);(2,7)(3,4)(9,10,14,18)(11,15,16,12)(13,17,22,24)(19,23,21,20)(25,27)(26,30)(28,34)(29,35)(31,33)(32,37)(36,40)(38,39);(1,3,6,4)(2,8,7,5)(9,10)(11,15)(12,16)(13,17)(14,18)(19,23)(20,21)(22,24)(25,28,31,26)(27,34,33,30)(29,36,38,32)(35,40,39,37);(1,4,6,3)(2,5,7,8)(9,12,14,15)(10,16,18,11)(13,20,22,23)(17,21,24,19)(25,29)(26,32)(27,35)(28,36)(30,37)(31,38)(33,39)(34,40);(1,5)(2,3)(4,7)(6,8)(9,13,14,22)(10,17,18,24)(11,19,16,21)(12,20,15,23)(25,30,31,34)(26,33,28,27)(29,37,38,40)(32,39,36,35)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=g,d^2=hi,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8706612);