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 | |
4511 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)(3,7)(4,5)(6,8)(9,10,14,15)(11,18,19,16)(12,13,17,23)(20,21,22,24)(25,26,31,29)(27,34,33,30)(28,36,38,32)(35,39,37,40);(2,5)(6,7)(10,15)(11,19)(12,17)(21,24)(25,27)(26,30)(28,37)(29,34)(31,33)(32,40)(35,38)(36,39);(1,2)(3,6)(4,5)(7,8)(11,19)(12,17)(13,23)(16,18)(26,29)(28,38)(30,34)(35,37);(1,3,4,8)(2,6,5,7)(9,11,14,19)(10,16,15,18)(12,22,17,20)(13,21,23,24)(25,28,31,38)(26,32,29,36)(27,35,33,37)(30,39,34,40);(9,12,14,17)(10,13,15,23)(11,20,19,22)(16,24,18,21)(25,29,31,26)(27,34,33,30)(28,32,38,36)(35,39,37,40);(9,13)(10,17)(11,21)(12,15)(14,23)(16,20)(18,22)(19,24)(25,30,31,34)(26,33,29,27)(28,39,38,40)(32,37,36,35)Polycyclic presentation:
2^9|a^2=gi,d^2=g,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8970477);