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)(12,20)(15,18)(19,23)(25,26,29,30)(27,33,35,31)(28,32,36,37)(34,39,40,38)(42,43)(44,51)(45,53)(54,55);(3,8)(5,7)(10,14)(11,17)(16,21)(19,23)(26,30)(28,36)(31,33)(34,40)(41,42,46,43)(44,47,51,49)(45,50,53,48)(52,55,56,54);(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,21)(22,23)(25,26)(27,31)(28,32)(29,30)(33,35)(34,38)(36,37)(39,40)(41,43,46,42)(44,49,51,47)(45,50,53,48)(52,55,56,54);(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)(28,34,36,40)(32,38,37,39)(41,44)(42,47)(43,49)(45,52)(46,51)(48,54)(50,55)(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)(26,32)(27,34)(29,36)(30,37)(31,38)(33,39)(35,40)(41,45)(42,48)(43,50)(44,52)(46,53)(47,54)(49,55)(51,56)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=gh,d^2=h,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, 5909039);