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 | |
4253 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)(9,10,14,15)(11,16,18,13)(12,20,22,17)(19,21,24,23)(25,26)(27,36)(28,39)(29,30)(31,35)(32,38)(33,40)(34,37);(1,3,5,8)(2,6,4,7)(25,27,31,38)(26,32,35,36)(28,37,40,29)(30,33,34,39);(1,2,5,4)(3,6,8,7)(10,15)(11,18)(17,20)(19,24)(25,28,31,40)(26,33,35,39)(27,29,38,37)(30,36,34,32);(1,4,5,2)(3,7,8,6)(9,11)(10,16)(12,19)(13,15)(14,18)(17,23)(20,21)(22,24)(25,28)(26,33)(27,37)(29,38)(30,36)(31,40)(32,34)(35,39);(9,12,14,22)(10,17,15,20)(11,19,18,24)(13,21,16,23)(25,29,31,37)(26,34,35,30)(27,28,38,40)(32,39,36,33);(9,13,14,16)(10,11,15,18)(12,21,22,23)(17,19,20,24)(25,30,31,34)(26,29,35,37)(27,33,38,39)(28,36,40,32)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=g,d^2=gi,e^2=h,f^2=h,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, 7803241);