Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #6132643 |
| 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:
(1,2,5,6)(3,4,7,8)(10,14)(15,17)(16,21)(22,23)(25,26)(27,31)(28,36)(29,30)(32,37)(33,34)(35,40)(38,39)(42,43)(47,49)(48,50)(54,55);(3,7)(4,8)(12,20)(16,21)(18,24)(22,23)(26,30)(27,33)(32,36)(35,39)(41,42,46,43)(44,49,52,47)(45,50,53,48)(51,54,56,55);(1,3)(2,4)(5,7)(6,8)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,26)(27,34)(28,36)(29,30)(31,33)(32,37)(35,38)(39,40)(41,43,46,42)(44,49,52,47)(45,48,53,50)(51,54,56,55);(1,4)(2,3)(5,8)(6,7)(9,11,13,19)(10,15,14,17)(12,18,20,24)(16,22,21,23)(25,27)(26,31)(28,35)(29,33)(30,34)(32,38)(36,40)(37,39)(41,44,46,52)(42,47,43,49)(45,51,53,56)(48,54,50,55);(9,12)(10,16)(11,18)(13,20)(14,21)(15,22)(17,23)(19,24)(25,28,29,37)(26,32,30,36)(27,35,33,39)(31,38,34,40)(41,45,46,53)(42,48,43,50)(44,51,52,56)(47,54,49,55)Polycyclic presentation:
2^9|a^2=i,b^2=fhi,c^2=fhi,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dfi,e^b=eg,d^c=dfg,e^c=ef62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6190594);