Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| 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:
(1,2,5,8)(3,6,4,7)(10,14)(15,17)(16,21)(22,23)(25,26)(27,31)(28,30)(29,32)(33,39)(34,35)(36,38)(37,40)(41,42,45,48)(43,49,50,46)(44,52,53,47)(51,54,56,55);(1,3,5,4)(2,6,8,7)(12,20)(16,21)(18,24)(22,23)(26,28)(31,35)(32,36)(39,40)(43,50)(44,53)(46,49)(47,52);(1,3,5,4)(2,6,8,7)(9,10,13,14)(11,17,19,15)(12,16,20,21)(18,23,24,22)(25,27,30,34)(26,31,28,35)(29,37,38,33)(32,40,36,39)(41,43)(42,46)(44,51)(45,50)(47,54)(48,49)(52,55)(53,56);(1,2)(3,6)(4,7)(5,8)(9,11,13,19)(10,15,14,17)(12,18,20,24)(16,22,21,23)(25,28,30,26)(27,31,34,35)(29,36,38,32)(33,39,37,40)(41,42)(43,49)(44,47)(45,48)(46,50)(51,55)(52,53)(54,56);(1,4,5,3)(2,7,8,6)(9,12)(10,16)(11,18)(13,20)(14,21)(15,22)(17,23)(19,24)(25,29,30,38)(26,32,28,36)(27,33,34,37)(31,39,35,40)(41,44)(42,47)(43,51)(45,53)(46,54)(48,52)(49,55)(50,56)Polycyclic presentation:
2^9|a^2=fi,b^2=fhi,c^2=gi,d^2=fgh,e^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5116886);