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 | |
4567 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,5)(3,7,8,6)(9,10,14,15)(11,18,16,12)(13,17,23,24)(19,22,21,20)(25,26)(27,31)(28,36)(29,34)(30,32)(33,38)(35,39)(37,40);(2,5)(6,7)(10,15)(11,16)(17,24)(19,21)(25,27,30,34)(26,31,32,29)(28,35,38,40)(33,37,36,39);(1,2)(3,6)(4,5)(7,8)(10,15)(11,16)(17,24)(19,21)(26,32)(27,34)(33,36)(35,40);(1,2)(3,6)(4,5)(7,8)(9,11,14,16)(10,12,15,18)(13,21,23,19)(17,22,24,20)(25,27)(26,31)(28,35)(29,32)(30,34)(33,37)(36,39)(38,40);(1,3)(2,6)(4,8)(5,7)(9,12)(10,16)(11,15)(13,22)(14,18)(17,19)(20,23)(21,24)(25,28,30,38)(26,33,32,36)(27,35,34,40)(29,39,31,37);(9,13,14,23)(10,17,15,24)(11,19,16,21)(12,20,18,22)(25,29,30,31)(26,34,32,27)(28,37,38,39)(33,35,36,40)Polycyclic presentation:
2^9|a^2=g,b^2=gh,d^2=hi,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8632021);