Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500044 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×D4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C42×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1374 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,6)(4,8)(5,7)(25,26,29,32,30,33,28,37)(27,34,35,40,36,39,38,31);(1,3,5,4)(2,6,7,8)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,27,30,36)(26,31,33,40)(28,35,29,38)(32,39,37,34);(2,7)(6,8)(9,11,13,19)(10,14,16,17)(12,20,21,18)(15,23,24,22)(25,28,30,29)(26,32,33,37)(27,35,36,38)(31,39,40,34);(9,12,13,21)(10,15,16,24)(11,18,19,20)(14,22,17,23)(26,33)(31,40)(32,37)(34,39);(1,4,5,3)(2,8,7,6)(10,16)(14,17)(15,24)(22,23)(26,33)(31,40)(32,37)(34,39)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=hi,d^2=h,e^2=gi,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ch,e^b=eh,f^b=fi,d^c=dh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6439644);