Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| 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:
(1,2,5,3)(4,7,8,6)(12,19)(16,21)(17,23)(22,24)(25,26,29,30)(27,31,34,37)(28,32,35,38)(33,39,36,40)(41,42,46,44)(43,49,50,47)(45,52,53,48)(51,54,56,55);(1,2,5,3)(4,6,8,7)(10,14)(15,20)(16,21)(22,24)(26,30)(28,35)(31,37)(33,36)(43,50)(45,53)(47,49)(48,52);(1,3,5,2)(4,7,8,6)(9,10,13,14)(11,15,18,20)(12,16,19,21)(17,22,23,24)(25,26)(27,31)(28,32)(29,30)(33,39)(34,37)(35,38)(36,40)(41,43)(42,47)(44,49)(45,51)(46,50)(48,54)(52,55)(53,56);(1,4)(2,6)(3,7)(5,8)(9,11,13,18)(10,15,14,20)(12,17,19,23)(16,22,21,24)(25,27,29,34)(26,31,30,37)(28,36,35,33)(32,40,38,39)(41,44,46,42)(43,47,50,49)(45,52,53,48)(51,54,56,55);(9,12)(10,16)(11,17)(13,19)(14,21)(15,22)(18,23)(20,24)(25,28,29,35)(26,32,30,38)(27,33,34,36)(31,39,37,40)(41,45)(42,48)(43,51)(44,52)(46,53)(47,54)(49,55)(50,56)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=ghi,d^2=h,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5700179);