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 | #50886 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4173 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)(16,18)(17,22)(23,24)(26,31)(27,34)(28,36)(38,40);(1,2,5,8)(3,4,7,6)(9,10)(11,16)(12,17)(13,18)(14,15)(19,23)(20,22)(21,24)(25,26)(27,32)(28,33)(29,34)(30,31)(35,40)(36,39)(37,38);(1,3)(2,6)(4,8)(5,7)(9,11,14,13)(10,16,15,18)(12,19,20,21)(17,23,22,24)(25,27,30,34)(26,32,31,29)(28,37,36,35)(33,38,39,40);(9,10)(11,16)(12,17)(13,18)(14,15)(19,23)(20,22)(21,24)(25,28,30,36)(26,33,31,39)(27,35,34,37)(29,38,32,40);(1,3)(2,4)(5,7)(6,8)(9,12)(10,17)(11,19)(13,21)(14,20)(15,22)(16,23)(18,24)(25,29)(26,27)(28,38)(30,32)(31,34)(33,35)(36,40)(37,39);(1,4)(2,7)(3,8)(5,6)(9,13,14,11)(10,18,15,16)(12,19,20,21)(17,23,22,24)(25,26)(27,32)(28,33)(29,34)(30,31)(35,40)(36,39)(37,38)Polycyclic presentation:
2^9|b^2=i,c^2=g,d^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9583822);