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 | #54930 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4288 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)(9,10,13,14)(11,17,18,15)(12,20,21,16)(19,22,24,23)(25,26,32,37)(27,36,31,33)(28,30,35,39)(29,38,40,34);(1,2)(3,6)(4,5)(7,8)(10,14)(15,17)(16,20)(22,23)(25,27,32,31)(26,33,37,36)(28,38,35,34)(29,39,40,30);(2,5)(6,8)(10,14)(11,18)(16,20)(19,24)(25,28)(26,30)(27,38)(29,33)(31,34)(32,35)(36,40)(37,39);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,22)(20,21)(23,24)(25,29,32,40)(26,34,37,38)(27,39,31,30)(28,36,35,33);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,23)(21,24)(25,30)(26,35)(27,29)(28,37)(31,40)(32,39)(33,34)(36,38);(1,3)(2,6)(4,7)(5,8)(9,12,13,21)(10,16,14,20)(11,19,18,24)(15,22,17,23)(25,31,32,27)(26,36,37,33)(28,34,35,38)(29,30,40,39)Polycyclic presentation:
2^9|a^2=g,b^2=gh,d^2=gh,f^2=g,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, 8032868);