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 | |
1258 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)(11,17)(12,19)(15,20)(16,21)(25,26,30,27)(28,31,36,33)(29,32,38,34)(35,39,37,40)(41,42,46,49)(43,44,51,47)(45,48,53,55)(50,52,56,54);(1,2,5,3)(4,6,8,7)(10,14)(15,20)(16,21)(22,24)(26,27)(31,33)(32,34)(39,40)(45,53)(48,55)(50,56)(52,54);(1,3,5,2)(4,7,8,6)(9,10,13,14)(11,15,17,20)(12,16,19,21)(18,22,23,24)(25,27,30,26)(28,31,36,33)(29,34,38,32)(35,39,37,40)(41,43,46,51)(42,47,49,44)(45,50,53,56)(48,54,55,52);(1,4)(2,6)(3,7)(5,8)(9,11,13,17)(10,15,14,20)(12,18,19,23)(16,22,21,24)(25,28)(26,31)(27,33)(29,37)(30,36)(32,40)(34,39)(35,38)(41,44,46,47)(42,43,49,51)(45,52,53,54)(48,50,55,56);(9,12)(10,16)(11,18)(13,19)(14,21)(15,22)(17,23)(20,24)(25,29,30,38)(26,32,27,34)(28,35,36,37)(31,39,33,40)(41,45,46,53)(42,48,49,55)(43,50,51,56)(44,52,47,54)Polycyclic presentation:
2^9|a^2=hi,b^2=fghi,c^2=fi,d^2=fgh,e^2=fg,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, 5699841);