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 | |
4484 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)(3,7)(4,5)(6,8)(9,10,14,15)(11,13,17,23)(12,19,20,16)(18,24,21,22)(25,26,30,35)(27,32,36,31)(28,37,39,33)(29,38,40,34);(2,5)(6,7)(10,15)(11,17)(12,20)(22,24)(25,26)(27,31)(28,33)(29,34)(30,35)(32,36)(37,39)(38,40);(1,2)(3,6)(4,5)(7,8)(11,17)(13,23)(18,21)(22,24)(25,27,30,36)(26,31,35,32)(28,34,39,38)(29,37,40,33);(1,3,4,8)(2,6,5,7)(9,11,14,17)(10,13,15,23)(12,21,20,18)(16,24,19,22)(25,27)(26,32)(28,38)(29,33)(30,36)(31,35)(34,39)(37,40);(9,12,14,20)(10,16,15,19)(11,18,17,21)(13,22,23,24)(25,28,30,39)(26,33,35,37)(27,34,36,38)(29,32,40,31);(9,13)(10,17)(11,15)(12,22)(14,23)(16,21)(18,19)(20,24)(25,29,30,40)(26,34,35,38)(27,33,36,37)(28,32,39,31)Polycyclic presentation:
2^9|a^2=gi,c^2=h,d^2=gh,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,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, 8842052);