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 | #38950 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4329 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10,14,17)(11,15,18,22)(12,13,16,21)(19,20,23,24)(25,26)(27,32)(28,30)(29,39)(31,35)(33,38)(34,36)(37,40);(1,3,5,8)(2,6,4,7)(9,11)(10,15)(12,19)(13,20)(14,18)(16,23)(17,22)(21,24)(25,27,31,37)(26,32,35,40)(28,34,38,39)(29,30,36,33);(1,2,5,4)(3,6,8,7)(11,18)(15,22)(19,23)(20,24)(25,28,31,38)(26,33,35,30)(27,34,37,39)(29,40,36,32);(1,2,5,4)(3,6,8,7)(9,11)(10,15)(12,19)(13,20)(14,18)(16,23)(17,22)(21,24)(25,29,31,36)(26,34,35,39)(27,30,37,33)(28,32,38,40);(1,2,5,4)(3,6,8,7)(9,12)(10,13)(11,19)(14,16)(15,20)(17,21)(18,23)(22,24)(25,30)(26,28)(27,36)(29,37)(31,33)(32,34)(35,38)(39,40);(1,4,5,2)(3,7,8,6)(9,13)(10,16)(11,20)(12,17)(14,21)(15,23)(18,24)(19,22)(25,26)(27,32)(28,33)(29,34)(30,38)(31,35)(36,39)(37,40)Polycyclic presentation:
2^9|a^2=gi,b^2=gh,c^2=gh,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dh,c^b=ci,d^c=dhi,f^e=fhi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7570328);