Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| 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)(11,17)(12,19)(15,20)(16,21)(25,26,30,27)(28,33,36,31)(29,32,37,34)(35,39,40,38)(41,42)(43,47)(44,46)(45,52)(48,51)(49,53)(50,55)(54,56);(10,14)(15,20)(16,21)(22,24)(26,27)(31,33)(32,34)(38,39)(41,43,46,51)(42,47,44,48)(45,50,53,56)(49,54,52,55);(1,2,4,6)(3,5,7,8)(9,10,13,14)(11,15,17,20)(12,16,19,21)(18,22,23,24)(25,27,30,26)(28,33,36,31)(29,32,37,34)(35,38,40,39)(41,43)(42,48)(44,47)(45,50)(46,51)(49,55)(52,54)(53,56);(1,3)(2,5)(4,7)(6,8)(9,11,13,17)(10,15,14,20)(12,18,19,23)(16,22,21,24)(25,28)(26,31)(27,33)(29,35)(30,36)(32,38)(34,39)(37,40)(41,44,46,42)(43,47,51,48)(45,52,53,49)(50,54,56,55);(9,12)(10,16)(11,18)(13,19)(14,21)(15,22)(17,23)(20,24)(25,29,30,37)(26,32,27,34)(28,35,36,40)(31,38,33,39)(41,45,46,53)(42,49,44,52)(43,50,51,56)(47,54,48,55)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=gi,d^2=fh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4543413);