Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1734 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,3)(4,7,8,6)(10,13)(14,16)(18,20)(19,23)(25,26)(27,32)(28,36)(29,34)(30,31)(33,39)(35,40)(37,38);(1,2,5,3)(4,7,8,6)(11,15)(14,16)(18,20)(22,24)(26,31)(27,34)(33,36)(35,37);(2,3)(6,7)(9,10,12,13)(11,14,15,16)(17,18,21,20)(19,24,23,22)(25,27,30,34)(26,32,31,29)(28,37,39,35)(33,40,36,38);(1,3,5,2)(4,7,8,6)(9,10)(11,14)(12,13)(15,16)(17,19,21,23)(18,22,20,24)(25,28)(26,33)(27,35)(29,40)(30,39)(31,36)(32,38)(34,37);(1,4,5,8)(2,6,3,7)(9,11)(10,14)(12,15)(13,16)(17,20,21,18)(19,24,23,22)(25,29,30,32)(26,34,31,27)(28,38,39,40)(33,35,36,37)Polycyclic presentation:
2^9|a^2=fgi,b^2=fgi,c^2=i,d^2=fh,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5712398);