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 | #52363 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4278 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,19)(12,17)(13,18)(14,15)(16,20)(21,24)(22,23)(25,26)(27,29)(28,36)(30,31)(32,34)(33,38)(35,39)(37,40);(1,2,5,4)(3,7,8,6)(11,20)(13,21)(16,19)(18,24)(26,31)(27,34)(33,36)(35,40);(1,3,5,8)(2,6,4,7)(9,11,15,20)(10,16,14,19)(12,13,23,21)(17,24,22,18)(25,27,30,34)(26,32,31,29)(28,35,38,40)(33,39,36,37);(3,8)(6,7)(9,12,15,23)(10,17,14,22)(11,21,20,13)(16,18,19,24)(25,28)(26,33)(27,35)(29,37)(30,38)(31,36)(32,39)(34,40);(1,3)(2,7)(4,6)(5,8)(9,13,15,21)(10,18,14,24)(11,12,20,23)(16,22,19,17)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,38)(37,40);(1,4,5,2)(3,6,8,7)(9,14,15,10)(11,16,20,19)(12,22,23,17)(13,24,21,18)(25,29,30,32)(26,34,31,27)(28,37,38,39)(33,40,36,35)Polycyclic presentation:
2^9|b^2=hi,c^2=i,d^2=gh,e^2=gh,f^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=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, 8935896);