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 | |
4299 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)(3,5)(4,7)(6,8)(11,17)(14,20)(18,23)(21,24)(25,26)(27,32)(28,30)(29,36)(31,35)(33,37)(34,38)(39,40);(2,3)(6,7)(9,10,13,16)(11,14,17,20)(12,15,19,22)(18,21,23,24)(25,27)(26,31)(28,32)(29,34)(30,35)(33,38)(36,39)(37,40);(1,2,5,3)(4,6,8,7)(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24)(25,27)(26,32)(28,31)(29,34)(30,35)(33,39)(36,38)(37,40);(1,3,5,2)(4,7,8,6)(9,10)(11,14)(12,15)(13,16)(17,20)(18,21)(19,22)(23,24)(25,28,30,26)(27,32,35,31)(29,36,37,33)(34,39,40,38);(1,4)(2,6)(3,7)(5,8)(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24)(25,27)(26,32)(28,31)(29,34)(30,35)(33,39)(36,38)(37,40);(9,12,13,19)(10,15,16,22)(11,18,17,23)(14,21,20,24)(25,29,30,37)(26,33,28,36)(27,34,35,40)(31,38,32,39)Polycyclic presentation:
2^9|b^2=hi,c^2=i,d^2=g,f^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, 8065983);