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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4546 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,20)(12,23)(13,24)(14,19)(15,17)(16,21)(18,22)(25,26)(27,28)(29,37)(30,34)(31,32)(33,39)(35,40)(36,38);(1,3,5,8)(2,6,4,7)(9,11)(10,16)(12,22)(13,19)(14,18)(15,21)(17,20)(23,24)(25,27,31,30)(26,28,32,34)(29,35,39,38)(33,36,37,40);(10,17)(11,21)(12,19)(18,24)(26,32)(28,34)(33,37)(36,40);(9,12,15,19)(10,14,17,23)(11,22,21,13)(16,18,20,24)(25,28,31,34)(26,27,32,30)(29,36,39,40)(33,35,37,38);(9,13,15,22)(10,18,17,24)(11,19,21,12)(14,20,23,16)(25,29,31,39)(26,33,32,37)(27,35,30,38)(28,36,34,40);(1,4,5,2)(3,7,8,6)(9,14)(10,19)(11,18)(12,17)(13,16)(15,23)(20,22)(21,24)(25,30,31,27)(26,34,32,28)(29,38,39,35)(33,40,37,36)Polycyclic presentation:
2^9|a^2=gi,b^2=gh,d^2=h,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7834311);