Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| 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 | |
1246 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)(12,19)(22,24)(25,26)(27,30)(28,33)(29,34)(31,35)(32,37)(36,38)(39,40)(41,42,46,43)(44,49,52,47)(45,50,53,48)(51,54,56,55);(1,2,4,6)(3,5,7,8)(10,14)(11,17)(16,21)(18,23)(26,27)(28,35)(29,37)(38,39)(42,43)(45,53)(47,49)(51,56);(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,27,30,26)(28,31,35,33)(29,34,37,32)(36,38,40,39)(41,43,46,42)(44,47,52,49)(45,50,53,48)(51,54,56,55);(1,3)(2,5)(4,7)(6,8)(9,11,13,17)(10,15,14,20)(12,18,19,23)(16,22,21,24)(25,28,30,35)(26,31,27,33)(29,36,37,40)(32,38,34,39)(41,44,46,52)(42,47,43,49)(45,51,53,56)(48,54,50,55);(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)(25,29)(26,32)(27,34)(28,36)(30,37)(31,38)(33,39)(35,40)(41,45,46,53)(42,48,43,50)(44,51,52,56)(47,54,49,55)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,c^2=f,d^2=h,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4384528);