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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4481 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,5)(4,8)(6,7)(9,10,13,14)(11,12,15,16)(18,23)(20,31)(21,33)(24,27)(28,41)(29,43)(38,45)(46,47);(2,7)(3,8)(10,14)(11,15)(18,23)(21,33)(24,27)(25,34)(29,43)(32,44)(36,39)(42,48);(1,3,6,8)(2,4,7,5)(9,11)(10,12)(13,15)(14,16)(17,18)(19,27)(20,25)(21,26)(22,23)(24,30)(28,39)(29,40)(31,34)(32,38)(33,35)(36,41)(37,43)(42,47)(44,45)(46,48);(1,4,6,5)(2,3,7,8)(9,12)(10,11)(13,16)(14,15)(17,19,22,30)(18,24,23,27)(20,28,31,41)(21,29,33,43)(25,36,34,39)(26,37,35,40)(32,42,44,48)(38,46,45,47);(1,2)(3,4)(5,8)(6,7)(17,20,22,31)(18,25,23,34)(19,28,30,41)(21,32,33,44)(24,36,27,39)(26,38,35,45)(29,42,43,48)(37,46,40,47);(1,5,6,4)(2,8,7,3)(17,21,22,33)(18,26,23,35)(19,29,30,43)(20,32,31,44)(24,37,27,40)(25,38,34,45)(28,42,41,48)(36,46,39,47)Polycyclic presentation:
2^9|a^2=hi,c^2=ghi,d^2=h,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8664573);