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 | #43912 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4365 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10,15,12)(11,19,21,16)(13,22,20,17)(14,18,24,23)(25,26)(27,30)(28,32)(29,33)(31,35)(34,38)(36,37)(39,40);(3,7)(5,8)(9,11,15,21)(10,16,12,19)(13,14,20,24)(17,18,22,23)(25,27)(26,30)(28,34)(29,35)(31,33)(32,38)(36,39)(37,40);(1,2)(3,5)(4,6)(7,8)(10,12)(16,19)(17,22)(18,23)(25,27,29,35)(26,30,33,31)(28,34,37,40)(32,38,36,39);(1,3,4,7)(2,5,6,8)(9,12,15,10)(11,16,21,19)(13,22,20,17)(14,18,24,23)(25,27)(26,31)(28,34)(29,35)(30,33)(32,39)(36,38)(37,40);(1,2)(3,5)(4,6)(7,8)(9,13)(10,17)(11,14)(12,22)(15,20)(16,18)(19,23)(21,24)(25,28)(26,32)(27,34)(29,37)(30,38)(31,39)(33,36)(35,40);(9,14)(10,18)(11,20)(12,23)(13,21)(15,24)(16,22)(17,19)(25,26)(27,30)(28,36)(29,33)(31,35)(32,37)(34,39)(38,40)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=i,d^2=h,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, 8646700);