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)(11,17)(16,21)(19,23)(25,26,30,27)(28,31,36,33)(29,34,37,32)(35,39,40,38)(43,49)(46,50)(51,55)(53,56);(3,8)(5,7)(10,14)(15,18)(16,21)(22,24)(26,27)(29,37)(31,33)(35,40)(41,42,45,48)(43,46,49,50)(44,47,52,54)(51,53,55,56);(9,10,13,14)(11,18,17,15)(12,16,20,21)(19,24,23,22)(25,27,30,26)(28,31,36,33)(29,34,37,32)(35,38,40,39)(41,43)(42,46)(44,51)(45,49)(47,53)(48,50)(52,55)(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)(25,28)(26,31)(27,33)(29,35)(30,36)(32,38)(34,39)(37,40)(41,42)(43,50)(44,47)(45,48)(46,49)(51,56)(52,54)(53,55);(1,2)(3,5)(4,6)(7,8)(9,12,13,20)(10,16,14,21)(11,19,17,23)(15,22,18,24)(25,29,30,37)(26,32,27,34)(28,35,36,40)(31,38,33,39)(41,44,45,52)(42,47,48,54)(43,51,49,55)(46,53,50,56)Polycyclic presentation:
2^9|a^2=hi,b^2=fgh,c^2=h,d^2=fh,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, 5908905);