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 | #2813 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4100 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)(3,8)(4,6)(5,7)(9,10,13,14)(11,17,18,15)(12,20,21,16)(19,22,24,23)(25,26)(27,32)(28,29)(30,40)(31,34)(33,38)(35,39)(36,37);(1,3,6,5)(2,7,4,8)(10,14)(11,18)(16,20)(19,24)(25,27,31,30)(26,32,34,40)(28,35,38,37)(29,39,33,36);(1,3,6,5)(2,7,4,8)(9,11,13,18)(10,15,14,17)(12,19,21,24)(16,22,20,23)(25,28)(26,33)(27,35)(29,34)(30,37)(31,38)(32,36)(39,40);(3,5)(7,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,22)(20,21)(23,24)(25,29,31,33)(26,28,34,38)(27,36,30,39)(32,37,40,35);(1,4,6,2)(3,8,5,7)(9,12,13,21)(10,16,14,20)(11,19,18,24)(15,22,17,23)(25,30,31,27)(26,32,34,40)(28,37,38,35)(29,39,33,36);(1,5,6,3)(2,7,4,8)Polycyclic presentation:
2^9|a^2=i,b^2=hi,c^2=g,d^2=gh,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8655125);