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 | |
1242 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,5,8)(3,6,4,7)(10,14)(11,17)(16,21)(19,23)(25,26)(27,33)(28,36)(29,30)(31,34)(32,37)(35,38)(39,40)(41,42,46,44)(43,49,50,47)(45,52,53,48)(51,54,56,55);(1,3,5,4)(2,6,8,7)(12,20)(16,21)(19,23)(22,24)(26,30)(27,34)(32,36)(35,40)(43,50)(45,53)(47,49)(48,52);(1,3,5,4)(2,6,8,7)(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,21)(22,23)(25,26)(27,33)(28,36)(29,30)(31,34)(32,37)(35,38)(39,40)(41,43,46,50)(42,47,44,49)(45,51,53,56)(48,54,52,55);(1,2)(3,6)(4,7)(5,8)(9,11)(10,15)(12,19)(13,17)(14,18)(16,22)(20,23)(21,24)(25,27)(26,31)(28,35)(29,34)(30,33)(32,38)(36,39)(37,40)(41,44,46,42)(43,47,50,49)(45,52,53,48)(51,54,56,55);(1,4,5,3)(2,7,8,6)(9,12,13,20)(10,16,14,21)(11,19,17,23)(15,22,18,24)(25,28,29,37)(26,32,30,36)(27,35,34,40)(31,38,33,39)(41,45,46,53)(42,48,44,52)(43,51,50,56)(47,54,49,55)Polycyclic presentation:
2^9|a^2=fgi,b^2=ghi,c^2=fgi,d^2=fh,e^2=fhi,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, 5116744);