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 | |
4529 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,5)(4,7)(6,8)(10,14)(11,18)(12,20)(22,23)(29,37)(32,39)(35,38)(36,40);(1,3)(2,4)(5,8)(6,7)(9,10)(11,15)(12,17)(13,14)(16,18)(19,22)(20,21)(23,24)(25,26)(27,34)(28,31)(29,32)(30,33)(35,40)(36,38)(37,39);(1,4,6,5)(2,7,8,3)(9,11,13,18)(10,15,14,16)(12,19,20,24)(17,22,21,23)(25,27,30,31)(26,28,33,34)(29,35,37,38)(32,36,39,40);(2,8)(4,5)(9,11)(10,16)(12,19)(13,18)(14,15)(17,23)(20,24)(21,22)(25,28,30,34)(26,31,33,27)(29,36,37,40)(32,38,39,35);(1,5,6,4)(2,3,8,7)(9,10)(11,15)(12,17)(13,14)(16,18)(19,22)(20,21)(23,24)(25,26)(27,28)(29,32)(30,33)(31,34)(35,36)(37,39)(38,40);(1,3)(2,4)(5,8)(6,7)(9,12)(10,17)(11,19)(13,20)(14,21)(15,22)(16,23)(18,24)(25,29)(26,32)(27,35)(28,36)(30,37)(31,38)(33,39)(34,40)Polycyclic presentation:
2^9|c^2=gi,d^2=i,e^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8068467);