Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| 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 | |
1278 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)(10,14)(11,17)(16,21)(18,23)(27,33)(28,35)(31,36)(32,37)(41,42,46,49)(43,47,51,44)(45,48,54,56)(50,55,53,52);(11,17)(15,20)(18,23)(22,24)(26,30)(31,36)(32,37)(38,40)(41,43,46,51)(42,44,49,47)(45,50,54,53)(48,52,56,55);(1,2,4,6)(3,5,7,8)(9,10,13,14)(11,15,17,20)(12,16,19,21)(18,22,23,24)(25,26,29,30)(27,31,33,36)(28,32,35,37)(34,38,39,40)(41,43)(42,47)(44,49)(45,53)(46,51)(48,52)(50,54)(55,56);(1,3)(2,5)(4,7)(6,8)(9,11,13,17)(10,15,14,20)(12,18,19,23)(16,22,21,24)(25,27,29,33)(26,31,30,36)(28,34,35,39)(32,38,37,40)(41,44,46,47)(42,43,49,51)(45,52,54,55)(48,50,56,53);(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)(25,28)(26,32)(27,34)(29,35)(30,37)(31,38)(33,39)(36,40)(41,45,46,54)(42,48,49,56)(43,50,51,53)(44,52,47,55)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=i,d^2=gh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4543471);