Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| 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 | |
1258 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)(12,19)(16,21)(17,23)(22,24)(25,26,29,32)(27,33,35,30)(28,31,36,38)(34,39,40,37)(41,42,46,43)(44,49,51,47)(45,50,53,48)(52,54,56,55);(1,3,5,2)(4,7,8,6)(10,14)(15,20)(16,21)(22,24)(28,36)(31,38)(34,40)(37,39)(42,43)(44,51)(48,50)(52,56);(9,10)(11,15)(12,16)(13,14)(17,22)(18,20)(19,21)(23,24)(25,26)(27,33)(28,31)(29,32)(30,35)(34,39)(36,38)(37,40)(41,43,46,42)(44,47,51,49)(45,48,53,50)(52,55,56,54);(1,4)(2,6)(3,7)(5,8)(9,11,13,18)(10,15,14,20)(12,17,19,23)(16,22,21,24)(25,27,29,35)(26,30,32,33)(28,34,36,40)(31,37,38,39)(41,44,46,51)(42,47,43,49)(45,52,53,56)(48,54,50,55);(9,12)(10,16)(11,17)(13,19)(14,21)(15,22)(18,23)(20,24)(25,28)(26,31)(27,34)(29,36)(30,37)(32,38)(33,39)(35,40)(41,45,46,53)(42,48,43,50)(44,52,51,56)(47,54,49,55)Polycyclic presentation:
2^9|a^2=i,b^2=fgi,c^2=f,d^2=gh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,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, 4974981);