Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| 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 | |
1246 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(3,7)(5,8)(10,14)(15,20)(16,21)(22,24)(25,26)(27,33)(28,30)(29,32)(31,35)(34,39)(36,37)(38,40)(41,42,46,47)(43,44,48,52)(45,53,54,49)(50,56,55,51);(1,2,4,6)(3,5,7,8)(11,17)(15,20)(18,23)(22,24)(26,28)(29,37)(31,33)(34,40)(42,47)(43,48)(45,54)(51,56);(9,10,13,14)(11,15,17,20)(12,16,19,21)(18,22,23,24)(25,27,30,35)(26,31,28,33)(29,34,37,40)(32,38,36,39)(41,43)(42,44)(45,50)(46,48)(47,52)(49,51)(53,56)(54,55);(1,3)(2,5)(4,7)(6,8)(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24)(25,28,30,26)(27,31,35,33)(29,36,37,32)(34,38,40,39)(41,44)(42,48)(43,47)(45,51)(46,52)(49,55)(50,53)(54,56);(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)(25,29)(26,32)(27,34)(28,36)(30,37)(31,38)(33,39)(35,40)(41,45,46,54)(42,49,47,53)(43,50,48,55)(44,51,52,56)Polycyclic presentation:
2^9|a^2=h,b^2=fhi,c^2=g,d^2=fh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4384566);