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 | |
1242 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)(3,7,8,5)(10,14)(15,18)(16,21)(22,24)(25,26)(27,31)(28,36)(29,30)(32,37)(33,35)(34,39)(38,40)(41,42,45,48)(43,46,51,49)(44,52,53,47)(50,55,56,54);(3,8)(5,7)(11,17)(15,18)(19,23)(22,24)(26,30)(28,37)(31,33)(34,40)(41,42,45,48)(43,49,51,46)(44,47,53,52)(50,55,56,54);(9,10,13,14)(11,18,17,15)(12,16,20,21)(19,24,23,22)(25,26)(27,33)(28,32)(29,30)(31,35)(34,39)(36,37)(38,40)(41,42)(43,49)(44,47)(45,48)(46,51)(50,55)(52,53)(54,56);(1,3)(2,5)(4,8)(6,7)(9,11,13,17)(10,15,14,18)(12,19,20,23)(16,22,21,24)(25,27,29,35)(26,31,30,33)(28,34,37,40)(32,38,36,39)(41,43)(42,46)(44,50)(45,51)(47,54)(48,49)(52,55)(53,56);(1,2)(3,5)(4,6)(7,8)(9,12,13,20)(10,16,14,21)(11,19,17,23)(15,22,18,24)(25,28,29,37)(26,32,30,36)(27,34,35,40)(31,38,33,39)(41,44,45,53)(42,47,48,52)(43,50,51,56)(46,54,49,55)Polycyclic presentation:
2^9|a^2=fhi,b^2=fh,c^2=fgh,d^2=gh,e^2=fg,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, 5909186);