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 | #49798 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4258 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,7)(6,8)(9,10)(11,16)(12,14)(13,20)(15,18)(17,21)(19,22)(23,24)(25,26,32,31)(27,36,30,33)(28,35,29,34)(37,39,38,40);(1,2)(3,6)(4,5)(7,8)(11,18)(15,16)(19,24)(22,23)(25,27,32,30)(26,33,31,36)(28,38,29,37)(34,40,35,39);(2,5)(6,8)(9,11,14,18)(10,15,12,16)(13,19,21,24)(17,22,20,23)(25,28,32,29)(26,34,31,35)(27,37,30,38)(33,39,36,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,15)(13,19)(14,18)(17,23)(20,22)(21,24)(25,29,32,28)(26,35,31,34)(27,38,30,37)(33,40,36,39);(1,2)(3,6)(4,5)(7,8)(9,12,14,10)(11,15,18,16)(13,20,21,17)(19,22,24,23)(25,30,32,27)(26,36,31,33)(28,38,29,37)(34,40,35,39);(1,3,4,7)(2,6,5,8)(9,13)(10,17)(11,19)(12,20)(14,21)(15,22)(16,23)(18,24)(25,31,32,26)(27,36,30,33)(28,35,29,34)(37,40,38,39)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=h,d^2=g,e^2=h,f^2=ghi,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, 8040057);