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 | |
4455 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10,14,15)(11,18,16,12)(13,17,22,23)(19,24,21,20)(25,26)(27,32)(28,34)(29,30)(31,33)(35,37)(36,38)(39,40);(3,7)(5,8)(10,15)(11,16)(17,23)(19,21)(25,27,31,36)(26,32,33,38)(28,37,39,29)(30,34,35,40);(1,2)(3,5)(4,6)(7,8)(10,15)(12,18)(17,23)(20,24)(26,33)(27,36)(29,37)(34,40);(1,3)(2,5)(4,7)(6,8)(9,11,14,16)(10,12,15,18)(13,21,22,19)(17,24,23,20)(25,28)(26,34)(27,29)(30,32)(31,39)(33,40)(35,38)(36,37);(1,2)(3,5)(4,6)(7,8)(9,12)(10,16)(11,15)(13,20)(14,18)(17,21)(19,23)(22,24)(25,29)(26,35)(27,28)(30,33)(31,37)(32,40)(34,38)(36,39);(9,13,14,22)(10,17,15,23)(11,19,16,21)(12,20,18,24)(25,30,31,35)(26,29,33,37)(27,34,36,40)(28,38,39,32)Polycyclic presentation:
2^9|a^2=hi,b^2=i,d^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8456861);