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 | #52004 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4716 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,7)(9,10)(11,16)(12,14)(13,20)(15,19)(17,21)(18,22)(23,24)(26,31)(27,33)(32,39)(34,35);(1,2,5,4)(3,7,8,6)(9,11,14,19)(10,15,12,16)(13,18,21,24)(17,22,20,23)(25,26)(27,29)(28,32)(30,31)(33,38)(34,36)(35,40)(37,39);(1,3,5,8)(2,6,4,7)(10,12)(13,21)(15,16)(18,24)(27,33)(29,38)(34,35)(36,40);(1,4,5,2)(3,7,8,6)(9,12,14,10)(11,16,19,15)(13,20,21,17)(18,23,24,22)(25,27)(26,29)(28,35)(30,33)(31,38)(32,40)(34,37)(36,39);(9,11)(10,16)(12,15)(13,18)(14,19)(17,23)(20,22)(21,24)(25,28,30,37)(26,32,31,39)(27,34,33,35)(29,36,38,40);(9,13,14,21)(10,17,12,20)(11,18,19,24)(15,22,16,23)(25,29)(26,27)(28,36)(30,38)(31,33)(32,34)(35,39)(37,40)Polycyclic presentation:
2^9|b^2=hi,c^2=i,d^2=hi,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8831951);