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 | #46612 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4417 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(10,15)(11,17)(16,23)(18,20)(25,26)(27,28)(29,37)(30,31)(32,35)(33,38)(34,39)(36,40);(1,2,5,4)(3,7,8,6)(9,10,14,15)(11,13,17,22)(12,16,19,23)(18,21,20,24)(26,31)(28,32)(33,37)(36,39);(1,3,5,8)(2,6,4,7)(11,17)(12,19)(13,22)(16,23)(25,26,30,31)(27,32,35,28)(29,33,38,37)(34,39,40,36);(3,8)(6,7)(9,11,14,17)(10,13,15,22)(12,20,19,18)(16,24,23,21)(25,27)(26,32)(28,31)(29,34)(30,35)(33,39)(36,37)(38,40);(1,3)(2,7)(4,6)(5,8)(9,12)(10,16)(11,18)(13,21)(14,19)(15,23)(17,20)(22,24)(25,28,30,32)(26,27,31,35)(29,36,38,39)(33,34,37,40);(1,4,5,2)(3,6,8,7)(9,13)(10,17)(11,15)(12,21)(14,22)(16,20)(18,23)(19,24)(25,29,30,38)(26,33,31,37)(27,34,35,40)(28,36,32,39)Polycyclic presentation:
2^9|b^2=gi,c^2=i,d^2=gh,e^2=h,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, 8935754);