Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5956507 |
| 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 | |
1256 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(2,5)(3,7)(9,10,13,16)(11,17,19,14)(12,20,21,15)(18,22,24,23)(25,26)(27,33)(28,36)(29,30)(31,34)(32,37)(35,38)(39,40)(42,43)(44,51)(45,53)(54,55);(2,5)(6,8)(26,30)(27,34)(28,37)(38,39)(41,42,46,43)(44,47,51,49)(45,50,53,48)(52,55,56,54);(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,21)(22,24)(25,27)(26,31)(28,35)(29,34)(30,33)(32,38)(36,39)(37,40)(41,43,46,42)(44,49,51,47)(45,50,53,48)(52,55,56,54);(1,2)(3,6)(4,5)(7,8)(9,11,13,19)(10,14,16,17)(12,18,21,24)(15,22,20,23)(25,26)(27,31)(28,32)(29,30)(33,34)(35,38)(36,37)(39,40)(41,44)(42,47)(43,49)(45,52)(46,51)(48,54)(50,55)(53,56);(1,3)(2,6)(4,7)(5,8)(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(25,28,29,37)(26,32,30,36)(27,35,34,40)(31,38,33,39)(41,45,46,53)(42,48,43,50)(44,52,51,56)(47,54,49,55)Polycyclic presentation:
2^9|a^2=fg,b^2=fh,c^2=fh,d^2=fg,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=ef,d^c=dfg62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5870450);