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 | #42057 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4371 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)(3,4)(5,8)(6,7)(9,10,15,12)(11,18,14,16)(13,17,22,20)(19,24,21,23)(25,26,31,33)(27,34,38,29)(28,30,40,32)(35,39,36,37);(1,3,7,8)(2,4,6,5)(9,11,15,14)(10,16,12,18)(13,21,22,19)(17,24,20,23)(25,27)(26,29)(28,35)(30,37)(31,38)(32,39)(33,34)(36,40);(1,2,7,6)(3,5,8,4)(25,28,31,40)(26,32,33,30)(27,35,38,36)(29,39,34,37);(1,4,7,5)(2,8,6,3)(9,12,15,10)(11,16,14,18)(13,20,22,17)(19,23,21,24)(25,28)(26,30)(27,36)(29,39)(31,40)(32,33)(34,37)(35,38);(1,5,7,4)(2,8,6,3)(9,13,15,22)(10,17,12,20)(11,19,14,21)(16,23,18,24)(25,29)(26,27)(28,39)(30,36)(31,34)(32,35)(33,38)(37,40);(1,6,7,2)(3,5,8,4)(9,14,15,11)(10,16,12,18)(13,19,22,21)(17,24,20,23)(25,30)(26,28)(27,37)(29,35)(31,32)(33,40)(34,36)(38,39)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=h,d^2=gh,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9842999);