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 | |
1258 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,5,3)(4,7,8,6)(11,17)(12,19)(15,20)(16,21)(25,26)(27,30)(28,33)(29,34)(31,35)(32,37)(36,38)(39,40)(41,42,45,48)(43,49,51,46)(44,47,52,54)(50,55,56,53);(1,3,5,2)(4,7,8,6)(10,14)(15,20)(16,21)(22,24)(26,27)(28,35)(29,37)(38,39)(44,52)(47,54)(50,56)(53,55);(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,27,30,26)(28,33,35,31)(29,32,37,34)(36,38,40,39)(41,42)(43,49)(44,47)(45,48)(46,51)(50,55)(52,54)(53,56);(1,4)(2,6)(3,7)(5,8)(9,11,13,17)(10,15,14,20)(12,18,19,23)(16,22,21,24)(25,28)(26,31)(27,33)(29,36)(30,35)(32,38)(34,39)(37,40)(41,43,45,51)(42,46,48,49)(44,50,52,56)(47,53,54,55);(9,12)(10,16)(11,18)(13,19)(14,21)(15,22)(17,23)(20,24)(25,29,30,37)(26,32,27,34)(28,36,35,40)(31,38,33,39)(41,44)(42,47)(43,50)(45,52)(46,53)(48,54)(49,55)(51,56)Polycyclic presentation:
2^9|a^2=hi,b^2=fghi,c^2=f,d^2=gh,e^2=f,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, 4976565);