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 | |
1238 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)(10,14)(11,17)(16,21)(19,23)(25,26,30,27)(28,33,35,31)(29,34,37,32)(36,38,40,39)(41,42,45,48)(43,46,49,53)(44,51,52,47)(50,56,55,54);(1,3,5,2)(4,7,8,6)(12,20)(16,21)(19,23)(22,24)(26,27)(28,35)(32,34)(36,40)(43,49)(46,53)(50,55)(54,56);(9,10,13,14)(11,18,17,15)(12,16,20,21)(19,24,23,22)(25,27,30,26)(28,31,35,33)(29,32,37,34)(36,39,40,38)(41,43,45,49)(42,46,48,53)(44,50,52,55)(47,54,51,56);(1,4,5,8)(2,6,3,7)(9,11,13,17)(10,15,14,18)(12,19,20,23)(16,22,21,24)(25,28,30,35)(26,31,27,33)(29,36,37,40)(32,38,34,39)(41,42)(43,46)(44,47)(45,48)(49,53)(50,54)(51,52)(55,56);(9,12)(10,16)(11,19)(13,20)(14,21)(15,22)(17,23)(18,24)(25,29,30,37)(26,32,27,34)(28,36,35,40)(31,38,33,39)(41,44,45,52)(42,47,48,51)(43,50,49,55)(46,54,53,56)Polycyclic presentation:
2^9|a^2=gi,b^2=ghi,c^2=gh,d^2=fhi,e^2=h,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, 5030291);