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 | #55126 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4221 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,12)(11,17,19,15)(13,16,21,20)(18,23,24,22)(26,31)(27,34)(28,36)(37,39);(1,2,5,6)(3,7,8,4)(10,12)(15,17)(16,20)(22,23)(25,26,30,31)(27,29,34,32)(28,33,36,38)(35,37,40,39);(2,6)(4,7)(13,21)(16,20)(18,24)(22,23)(25,27,30,34)(26,32,31,29)(28,35,36,40)(33,39,38,37);(1,3,5,8)(2,7,6,4)(9,11,14,19)(10,15,12,17)(13,18,21,24)(16,22,20,23)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,38)(37,40);(1,3)(2,4)(5,8)(6,7)(9,12,14,10)(11,15,19,17)(13,20,21,16)(18,22,24,23)(25,28)(26,33)(27,35)(29,37)(30,36)(31,38)(32,39)(34,40);(1,4)(2,3)(5,7)(6,8)(9,13,14,21)(10,16,12,20)(11,18,19,24)(15,22,17,23)(25,29,30,32)(26,34,31,27)(28,37,36,39)(33,40,38,35)Polycyclic presentation:
2^9|a^2=gh,b^2=i,c^2=h,d^2=gi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8265875);