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 | |
4290 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(10,15)(11,17)(13,21)(19,24)(25,26)(27,32)(28,29)(30,40)(31,34)(33,38)(35,39)(36,37);(1,2)(3,7)(4,5)(6,8)(9,10,14,15)(11,12,17,20)(13,22,21,16)(18,24,23,19)(25,27,31,30)(26,32,34,40)(28,35,38,37)(29,39,33,36);(2,5)(6,7)(9,11,14,17)(10,12,15,20)(13,18,21,23)(16,19,22,24)(25,28)(26,33)(27,35)(29,34)(30,37)(31,38)(32,36)(39,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,12)(13,18)(14,17)(15,20)(16,19)(21,23)(22,24);(1,2)(3,6)(4,5)(7,8)(9,12)(10,11)(13,19)(14,20)(15,17)(16,18)(21,24)(22,23)(25,29,31,33)(26,28,34,38)(27,36,30,39)(32,37,40,35);(1,3)(2,6)(4,8)(5,7)(9,13,14,21)(10,16,15,22)(11,18,17,23)(12,19,20,24)(25,30,31,27)(26,32,34,40)(28,37,38,35)(29,39,33,36)Polycyclic presentation:
2^9|b^2=h,c^2=g,e^2=gh,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, 8668037);