Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4932388 |
| 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 | |
1504 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,6,7)(3,4,5,8)(10,14)(11,15)(17,18,22,24)(19,23,21,20)(25,26,29,32)(27,33,34,30)(28,36,37,31)(35,38,40,39);(1,3,6,5)(2,4,7,8)(9,10,13,14)(11,12,15,16)(17,19,22,21)(18,23,24,20)(27,34)(30,33)(35,40)(38,39);(2,7)(4,8)(10,14)(11,15)(17,20,22,23)(18,21,24,19)(25,27)(26,30)(28,35)(29,34)(31,38)(32,33)(36,39)(37,40);(1,4,6,8)(2,3,7,5)(9,11)(10,12)(13,15)(14,16)(17,20)(18,19)(21,24)(22,23)(25,26)(27,33)(28,31)(29,32)(30,34)(35,39)(36,37)(38,40);(1,5,6,3)(2,8,7,4)(9,12)(10,11)(13,16)(14,15)(17,21,22,19)(18,20,24,23)(25,28,29,37)(26,31,32,36)(27,35,34,40)(30,38,33,39)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=fgi,d^2=hi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4908960);