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 | #34741 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4375 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10)(11,14)(12,20)(13,15)(16,18)(17,21)(19,23)(22,24)(25,26)(27,34)(28,36)(29,31)(30,32)(33,38)(35,39)(37,40);(3,7)(5,8)(9,11)(10,14)(12,19)(13,18)(15,16)(17,22)(20,23)(21,24)(25,27,31,30)(26,32,29,34)(28,37,38,35)(33,40,36,39);(1,2)(3,5)(4,6)(7,8)(10,15)(11,18)(17,20)(19,24)(26,29)(28,38)(32,34)(35,37);(1,3)(2,5)(4,7)(6,8)(9,10)(11,16)(12,17)(13,15)(14,18)(19,23)(20,21)(22,24)(25,28)(26,33)(27,35)(29,36)(30,37)(31,38)(32,39)(34,40);(1,2)(3,5)(4,6)(7,8)(9,11)(10,16)(12,19)(13,18)(14,15)(17,23)(20,22)(21,24)(25,29,31,26)(27,34,30,32)(28,36,38,33)(35,40,37,39);(9,12,13,21)(10,17,15,20)(11,19,18,24)(14,22,16,23)(25,30,31,27)(26,34,29,32)(28,37,38,35)(33,40,36,39)Polycyclic presentation:
2^9|a^2=ghi,b^2=g,e^2=g,f^2=gi,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^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8040131);