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)(15,17)(16,21)(22,23)(25,26)(27,30)(28,33)(29,34)(31,35)(32,37)(36,38)(39,40)(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)(18,24)(22,23)(26,27)(28,35)(29,37)(38,39)(43,49)(46,53)(50,55)(54,56);(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,27,30,26)(28,33,35,31)(29,32,37,34)(36,38,40,39)(41,43,45,49)(42,46,48,53)(44,50,52,55)(47,54,51,56);(1,4,5,8)(2,6,3,7)(9,11,13,19)(10,15,14,17)(12,18,20,24)(16,22,21,23)(25,28)(26,31)(27,33)(29,36)(30,35)(32,38)(34,39)(37,40)(41,42)(43,46)(44,47)(45,48)(49,53)(50,54)(51,52)(55,56);(9,12,13,20)(10,16,14,21)(11,18,19,24)(15,22,17,23)(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=fi,b^2=hi,c^2=h,d^2=fghi,e^2=fgh,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, 5030476);