Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| 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 | |
1246 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(3,7)(5,8)(12,19)(16,21)(17,23)(22,24)(25,26)(27,31)(28,35)(29,30)(32,37)(33,40)(34,38)(36,39)(41,42,45,46)(43,47,49,54)(44,51,53,48)(50,56,52,55);(1,2,4,6)(3,5,7,8)(10,14)(15,20)(16,21)(22,24)(26,30)(31,38)(32,35)(39,40)(42,46)(43,49)(48,51)(50,52);(9,10,13,14)(11,15,18,20)(12,16,19,21)(17,22,23,24)(25,26)(27,31)(28,32)(29,30)(33,39)(34,38)(35,37)(36,40)(41,42)(43,47)(44,48)(45,46)(49,54)(50,55)(51,53)(52,56);(1,3)(2,5)(4,7)(6,8)(9,11,13,18)(10,15,14,20)(12,17,19,23)(16,22,21,24)(25,27,29,34)(26,31,30,38)(28,36,37,33)(32,40,35,39)(41,43)(42,47)(44,52)(45,49)(46,54)(48,56)(50,53)(51,55);(9,12)(10,16)(11,17)(13,19)(14,21)(15,22)(18,23)(20,24)(25,28,29,37)(26,32,30,35)(27,33,34,36)(31,39,38,40)(41,44)(42,48)(43,50)(45,53)(46,51)(47,55)(49,52)(54,56)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=fh,d^2=gh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4780248);