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)(27,30)(28,33)(29,32)(31,36)(34,37)(35,40)(38,39)(41,42,45,48)(43,49,51,46)(44,47,52,54)(50,55,56,53);(1,2,5,3)(4,6,8,7)(10,14)(15,20)(16,21)(22,24)(26,27)(29,37)(31,33)(35,38)(44,52)(47,54)(50,56)(53,55);(1,3,5,2)(4,7,8,6)(9,10,13,14)(11,15,18,20)(12,16,19,21)(17,22,23,24)(25,27,30,26)(28,33,36,31)(29,34,37,32)(35,40,38,39)(41,43,45,51)(42,46,48,49)(44,50,52,56)(47,53,54,55);(1,4)(2,6)(3,7)(5,8)(9,11,13,18)(10,15,14,20)(12,17,19,23)(16,22,21,24)(25,28)(26,31)(27,33)(29,38)(30,36)(32,40)(34,39)(35,37)(41,42)(43,49)(44,47)(45,48)(46,51)(50,55)(52,54)(53,56);(9,12)(10,16)(11,17)(13,19)(14,21)(15,22)(18,23)(20,24)(25,29,30,37)(26,32,27,34)(28,35,36,38)(31,39,33,40)(41,44,45,52)(42,47,48,54)(43,50,51,56)(46,53,49,55)Polycyclic presentation:
2^9|a^2=fi,b^2=gi,c^2=fhi,d^2=fh,e^2=g,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, 5700592);