Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1438 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,3)(4,8)(9,10,13,16)(11,14,12,15)(17,18,21,20)(19,22,23,24)(25,26,31,34)(27,29,28,32)(30,33,39,40)(35,37,36,38);(2,3)(6,7)(19,23)(22,24)(25,27,31,28)(26,32,34,29)(30,35,39,36)(33,38,40,37);(1,2,5,3)(4,6,8,7)(9,11,13,12)(10,14,16,15)(17,19)(18,22)(20,24)(21,23)(25,28,31,27)(26,32,34,29)(30,35,39,36)(33,37,40,38);(1,3,5,2)(4,7,8,6)(9,10)(11,14)(12,15)(13,16)(17,18,21,20)(19,22,23,24)(25,29,31,32)(26,27,34,28)(30,38,39,37)(33,36,40,35);(1,4,5,8)(2,6,3,7)(9,12,13,11)(10,15,16,14)(17,20,21,18)(19,24,23,22)(25,30)(26,33)(27,35)(28,36)(29,37)(31,39)(32,38)(34,40)Polycyclic presentation:
2^9|a^2=hi,b^2=f,c^2=i,d^2=fh,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3627523);