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 | |
4694 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,12)(13,17)(14,15)(16,18)(19,20)(21,22)(23,24)(25,26)(27,30)(28,38)(29,40)(31,35)(32,37)(33,39)(34,36);(1,2)(3,6)(4,7)(5,8)(9,10)(11,18)(12,16)(13,17)(14,15)(19,22)(20,21)(23,24)(25,27)(26,32)(28,29)(30,35)(31,37)(33,40)(34,38)(36,39);(1,3)(2,5)(4,8)(6,7)(10,15)(11,16)(17,24)(19,21)(25,28,31,39)(26,33,35,38)(27,29,37,36)(30,34,32,40);(3,8)(5,6)(9,11,14,16)(10,12,15,18)(13,21,23,19)(17,22,24,20)(25,28)(26,33)(27,36)(29,37)(30,40)(31,39)(32,34)(35,38);(1,2)(3,5)(4,7)(6,8)(9,12)(10,16)(11,15)(13,22)(14,18)(17,19)(20,23)(21,24)(25,29,31,36)(26,34,35,40)(27,28,37,39)(30,33,32,38);(1,3)(2,6)(4,8)(5,7)(9,13,14,23)(10,17,15,24)(11,19,16,21)(12,20,18,22)(25,30,31,32)(26,27,35,37)(28,34,39,40)(29,38,36,33)Polycyclic presentation:
2^9|c^2=g,d^2=gh,e^2=g,f^2=h,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, 8611378);