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 | |
4402 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,19)(12,22)(13,16)(14,20)(15,17)(18,23)(21,24)(25,26)(27,31)(28,36)(29,30)(32,38)(33,35)(34,40)(37,39);(1,2,5,4)(3,6,7,8)(9,11,15,13)(10,16,17,19)(12,20,23,21)(14,22,24,18)(25,26)(27,33)(28,32)(29,30)(31,35)(34,40)(36,38)(37,39);(2,4)(6,8)(10,17)(14,24)(16,19)(18,22)(26,30)(31,33)(32,36)(39,40);(3,7)(6,8)(9,12,15,23)(10,18,17,22)(11,21,13,20)(14,16,24,19)(25,27)(26,31)(28,37)(29,35)(30,33)(32,40)(34,38)(36,39);(1,3)(2,6)(4,8)(5,7)(9,13,15,11)(10,19,17,16)(12,21,23,20)(14,18,24,22)(25,28)(26,32)(27,34)(29,38)(30,36)(31,39)(33,40)(35,37);(1,4,5,2)(3,8,7,6)(9,14,15,24)(10,20,17,21)(11,22,13,18)(12,16,23,19)(25,26)(27,31)(28,32)(29,30)(33,35)(34,39)(36,38)(37,40)Polycyclic presentation:
2^9|b^2=hi,d^2=g,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8232866);