Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| 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 | |
1254 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)(9,10,14,11)(12,17,19,15)(13,16,21,18)(20,23,24,22)(25,26,29,32)(27,30,33,37)(28,35,36,31)(34,40,39,38)(42,46)(43,49)(48,52)(51,55);(10,11)(12,19)(16,18)(20,24)(27,33)(30,37)(34,39)(38,40)(41,42,45,46)(43,50,49,47)(44,52,53,48)(51,54,55,56);(1,2,4,6)(3,5,7,8)(9,11,14,10)(12,15,19,17)(13,16,21,18)(20,23,24,22)(25,27,29,33)(26,30,32,37)(28,34,36,39)(31,38,35,40)(41,42)(43,47)(44,52)(45,46)(48,53)(49,50)(51,56)(54,55);(1,3,4,7)(2,5,6,8)(9,12,14,19)(10,15,11,17)(13,20,21,24)(16,22,18,23)(25,26)(27,30)(28,31)(29,32)(33,37)(34,38)(35,36)(39,40)(41,43,45,49)(42,47,46,50)(44,51,53,55)(48,54,52,56);(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,28,29,36)(26,31,32,35)(27,34,33,39)(30,38,37,40)(41,44,45,53)(42,48,46,52)(43,51,49,55)(47,54,50,56)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=gi,d^2=fhi,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efh,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, 5243475);