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 | |
1240 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,4,6)(3,5,7,8)(12,19)(16,21)(17,23)(22,24)(25,26)(27,30)(28,31)(29,34)(32,37)(33,35)(36,39)(38,40)(42,46)(47,49)(48,54)(55,56);(10,14)(15,20)(16,21)(22,24)(26,27)(28,35)(29,37)(38,39)(41,42,45,46)(43,47,51,49)(44,48,53,54)(50,55,52,56);(9,10,13,14)(11,15,18,20)(12,16,19,21)(17,22,23,24)(25,27,30,26)(28,33,35,31)(29,32,37,34)(36,38,40,39)(41,43)(42,47)(44,52)(45,51)(46,49)(48,56)(50,53)(54,55);(1,2)(3,5)(4,6)(7,8)(9,11,13,18)(10,15,14,20)(12,17,19,23)(16,22,21,24)(25,28,30,35)(26,31,27,33)(29,36,37,40)(32,38,34,39)(41,42)(43,49)(44,48)(45,46)(47,51)(50,56)(52,55)(53,54);(1,3,4,7)(2,5,6,8)(9,12,13,19)(10,16,14,21)(11,17,18,23)(15,22,20,24)(25,29)(26,32)(27,34)(28,36)(30,37)(31,38)(33,39)(35,40)(41,44,45,53)(42,48,46,54)(43,50,51,52)(47,55,49,56)Polycyclic presentation:
2^9|a^2=fgi,b^2=fg,c^2=h,d^2=h,e^2=ghi,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, 5093166);