Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| 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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1311 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
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)(27,30,35,33)(28,31,36,38)(34,37,40,39)(41,42,46,47)(43,44,48,52)(45,49,54,55)(50,51,56,53);(1,2,5,3)(4,6,8,7)(10,14)(15,20)(16,21)(22,24)(28,36)(31,38)(34,40)(37,39)(42,47)(43,48)(49,55)(50,56);(1,3,5,2)(4,7,8,6)(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)(42,44)(45,50)(46,48)(47,52)(49,51)(53,55)(54,56);(1,4,5,8)(2,6,3,7)(9,11,13,18)(10,15,14,20)(12,17,19,23)(16,22,21,24)(25,27)(26,30)(28,34)(29,35)(31,37)(32,33)(36,40)(38,39)(41,44)(42,48)(43,47)(45,53)(46,52)(49,50)(51,54)(55,56);(9,12,13,19)(10,16,14,21)(11,17,18,23)(15,22,20,24)(25,28)(26,31)(27,34)(29,36)(30,37)(32,38)(33,39)(35,40)(41,45)(42,49)(43,50)(44,51)(46,54)(47,55)(48,56)(52,53)Polycyclic presentation:
2^9|a^2=fgi,b^2=i,c^2=i,d^2=fhi,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5854364);