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 |
| prime radical | #392 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4126 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,18)(11,19,22,13)(12,17,14,16)(20,23,21,24)(25,26,31,27)(28,34,30,32)(29,33,38,35)(36,40,37,39);(1,2)(3,8)(4,5)(6,7)(9,11,15,22)(10,13,18,19)(12,20,14,21)(16,23,17,24)(25,27,31,26)(28,34,30,32)(29,33,38,35)(36,39,37,40);(2,7)(3,4)(9,11)(10,13)(12,20)(14,21)(15,22)(16,23)(17,24)(18,19)(25,28,31,30)(26,32,27,34)(29,36,38,37)(33,39,35,40);(1,3,6,4)(2,8,7,5)(9,12,15,14)(10,16,18,17)(11,20,22,21)(13,23,19,24);(1,4,6,3)(2,5,7,8)(9,13)(10,11)(12,23)(14,24)(15,19)(16,20)(17,21)(18,22)(25,29)(26,33)(27,35)(28,36)(30,37)(31,38)(32,39)(34,40);(1,5)(2,3)(4,7)(6,8)(9,14,15,12)(10,17,18,16)(11,21,22,20)(13,24,19,23)(25,30,31,28)(26,34,27,32)(29,37,38,36)(33,40,35,39)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=gh,d^2=i,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8706433);