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 | |
1598 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,5)(3,7,8,6)(9,10)(11,12)(13,14)(15,16)(18,19)(20,24)(25,26)(27,32)(28,33)(29,34)(30,31)(35,40)(36,38)(37,39);(1,2)(3,6)(4,5)(7,8)(9,10,13,14)(11,12,15,16)(17,18,21,19)(20,22,24,23)(26,31)(28,37)(32,38)(34,35);(2,5)(6,7)(10,14)(12,16)(17,18,21,19)(20,23,24,22)(25,27)(26,32)(28,35)(29,39)(30,36)(31,38)(33,40)(34,37);(1,3,4,8)(2,6,5,7)(9,11)(10,12)(13,15)(14,16)(17,19,21,18)(20,22,24,23)(25,28,30,37)(26,33,31,39)(27,35,36,34)(29,32,40,38);(1,2)(3,7)(4,5)(6,8)(9,12)(10,15)(11,14)(13,16)(17,20)(18,22)(19,23)(21,24)(25,29,30,40)(26,34,31,35)(27,33,36,39)(28,38,37,32)Polycyclic presentation:
2^9|a^2=ghi,b^2=hi,c^2=gh,d^2=fgi,e^2=fg,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, 3424737);