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 | #46032 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4316 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(25,26,32,36)(27,37,39,33)(28,30,34,29)(31,35,40,38);(3,7)(6,8)(9,10)(11,15)(12,19)(13,14)(16,20)(17,23)(18,21)(22,24)(25,27,32,39)(26,33,36,37)(28,31,34,40)(29,35,30,38);(1,2)(3,6)(4,5)(7,8)(10,14)(15,21)(16,19)(22,23)(25,28)(26,29)(27,31)(30,36)(32,34)(33,35)(37,38)(39,40);(1,3,4,7)(2,6,5,8)(9,11,13,18)(10,15,14,21)(12,17,20,24)(16,22,19,23)(25,29,32,30)(26,34,36,28)(27,38,39,35)(31,37,40,33);(1,2)(3,6)(4,5)(7,8)(9,12)(10,16)(11,17)(13,20)(14,19)(15,22)(18,24)(21,23)(25,30,32,29)(26,28,36,34)(27,35,39,38)(31,33,40,37);(9,10)(11,15)(12,16)(13,14)(17,22)(18,21)(19,20)(23,24)(25,31)(26,35)(27,34)(28,39)(29,37)(30,33)(32,40)(36,38)Polycyclic presentation:
2^9|a^2=g,b^2=g,d^2=hi,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8113348);