Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| 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 | |
1370 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(6,8)(9,10,14,15)(11,16,13,12)(19,20)(23,24)(25,26)(27,31)(28,30)(29,36)(32,35)(33,38)(34,40)(37,39);(9,11)(10,12)(13,14)(15,16)(18,22)(23,24)(25,27,30,35)(26,31,28,32)(29,37,38,34)(33,40,36,39);(9,12,14,16)(10,11,15,13)(17,18)(19,23)(20,24)(21,22)(25,27)(26,32)(28,31)(29,34)(30,35)(33,40)(36,39)(37,38);(1,2)(3,6)(4,5)(7,8)(9,13,14,11)(10,12,15,16)(17,19,21,20)(18,23,22,24)(25,28,30,26)(27,32,35,31)(29,36,38,33)(34,40,37,39);(1,3,4,7)(2,6,5,8)(9,12)(10,11)(13,15)(14,16)(17,20,21,19)(18,24,22,23)(25,29)(26,33)(27,34)(28,36)(30,38)(31,39)(32,40)(35,37)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,d^2=h,e^2=gi,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5258656);