Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #6132643 |
| 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 | |
1240 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,4,6)(3,5,7,8)(10,14)(15,17)(16,21)(22,23)(25,26,30,27)(28,31,35,33)(29,34,37,32)(36,39,40,38)(41,42)(43,49)(44,52)(45,51)(46,47)(48,54)(50,53)(55,56);(1,3,4,7)(2,5,6,8)(12,20)(16,21)(18,24)(22,23)(26,27)(28,35)(32,34)(36,40)(42,47)(45,49)(48,52)(53,55);(1,3)(2,5)(4,7)(6,8)(9,10,13,14)(11,17,19,15)(12,16,20,21)(18,23,24,22)(25,27,30,26)(28,31,35,33)(29,32,37,34)(36,39,40,38)(41,43,46,51)(42,45,47,49)(44,50,54,56)(48,53,52,55);(9,11,13,19)(10,15,14,17)(12,18,20,24)(16,22,21,23)(25,28)(26,31)(27,33)(29,36)(30,35)(32,38)(34,39)(37,40)(41,44,46,54)(42,48,47,52)(43,50,51,56)(45,53,49,55);(1,2)(3,5)(4,6)(7,8)(9,12,13,20)(10,16,14,21)(11,18,19,24)(15,22,17,23)(25,29,30,37)(26,32,27,34)(28,36,35,40)(31,38,33,39)(41,45)(42,43)(44,53)(46,49)(47,51)(48,50)(52,56)(54,55)Polycyclic presentation:
2^9|a^2=h,b^2=fhi,c^2=fgi,d^2=g,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dfi,e^b=eg,d^c=dfg,e^c=ef62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6176891);