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 | #51700 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4303 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,17)(12,16)(13,14)(15,18)(19,23)(20,21)(22,24)(25,26,31,29)(27,28,34,32)(30,37,38,33)(35,39,40,36);(1,2)(3,8)(4,5)(6,7)(10,14)(15,17)(16,21)(22,23)(26,29)(27,34)(33,37)(35,40);(2,7)(3,4)(11,18)(12,20)(15,17)(16,21)(25,27,31,34)(26,32,29,28)(30,35,38,40)(33,39,37,36);(1,3,6,4)(2,8,7,5)(9,11,13,18)(10,15,14,17)(12,19,20,24)(16,22,21,23)(25,28)(26,27)(29,34)(30,36)(31,32)(33,35)(37,40)(38,39);(1,4,6,3)(2,5,7,8)(9,12)(10,16)(11,19)(13,20)(14,21)(15,22)(17,23)(18,24)(25,29,31,26)(27,28,34,32)(30,37,38,33)(35,36,40,39);(1,5)(2,3)(4,7)(6,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,22)(20,21)(23,24)(25,30,31,38)(26,33,29,37)(27,35,34,40)(28,36,32,39)Polycyclic presentation:
2^9|a^2=h,c^2=h,d^2=gi,e^2=i,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, 8706341);