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 | |
1242 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,5,3)(4,7,8,6)(10,14)(11,17)(16,21)(19,23)(25,26)(27,31)(28,32)(29,30)(33,34)(35,38)(36,37)(39,40)(41,42,45,48)(43,49,50,46)(44,52,53,47)(51,54,56,55);(1,3,5,2)(4,7,8,6)(12,20)(16,21)(19,23)(22,24)(26,30)(27,33)(32,36)(35,39)(43,50)(44,53)(46,49)(47,52);(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,21)(22,23)(25,26)(27,34)(28,36)(29,30)(31,33)(32,37)(35,38)(39,40)(41,43,45,50)(42,46,48,49)(44,51,53,56)(47,54,52,55);(1,4)(2,6)(3,7)(5,8)(9,11)(10,15)(12,19)(13,17)(14,18)(16,22)(20,23)(21,24)(25,27,29,33)(26,31,30,34)(28,35,37,39)(32,38,36,40)(41,42)(43,49)(44,47)(45,48)(46,50)(51,55)(52,53)(54,56);(9,12)(10,16)(11,19)(13,20)(14,21)(15,22)(17,23)(18,24)(25,28)(26,32)(27,35)(29,37)(30,36)(31,38)(33,39)(34,40)(41,44,45,53)(42,47,48,52)(43,51,50,56)(46,54,49,55)Polycyclic presentation:
2^9|a^2=fghi,b^2=gi,c^2=fh,d^2=f,e^2=fh,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, 5150680);