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 | |
1260 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)(11,17)(12,19)(14,20)(15,22)(25,26,29,30)(27,31,33,37)(28,35,36,32)(34,40,39,38)(41,42)(43,47)(44,46)(45,52)(48,53)(49,55)(50,51)(54,56);(1,2,4,6)(3,8,7,5)(11,17)(14,20)(18,23)(21,24)(26,30)(27,33)(28,36)(38,40)(42,44)(45,53)(47,51)(49,56);(1,2)(3,5)(4,6)(7,8)(9,10,13,16)(11,14,17,20)(12,15,19,22)(18,21,23,24)(25,26)(27,31)(28,32)(29,30)(33,37)(34,38)(35,36)(39,40)(41,43)(42,47)(44,51)(45,49)(46,50)(48,54)(52,55)(53,56);(1,3)(2,5)(4,7)(6,8)(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24)(25,27,29,33)(26,31,30,37)(28,34,36,39)(32,38,35,40)(41,44,46,42)(43,47,50,51)(45,52,53,48)(49,54,56,55);(9,12)(10,15)(11,18)(13,19)(14,21)(16,22)(17,23)(20,24)(25,28,29,36)(26,32,30,35)(27,34,33,39)(31,38,37,40)(41,45)(42,48)(43,49)(44,52)(46,53)(47,54)(50,56)(51,55)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=fhi,d^2=f,e^2=g,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, 5926390);