Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| 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 | |
1238 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(3,7)(5,8)(10,14)(12,19)(15,20)(18,23)(25,26,29,32)(27,30,33,36)(28,31,35,38)(34,37,39,40)(41,42)(43,49)(44,51)(45,53)(46,47)(48,55)(50,52)(54,56);(1,2,4,6)(3,5,7,8)(10,14)(11,17)(16,21)(18,23)(27,33)(30,36)(34,39)(37,40)(42,47)(44,49)(48,53)(52,56);(9,10,13,14)(11,15,17,20)(12,16,19,21)(18,22,23,24)(25,26)(27,30)(28,31)(29,32)(33,36)(34,37)(35,38)(39,40)(41,43)(42,44)(45,54)(46,51)(47,49)(48,56)(50,55)(52,53);(1,3,4,7)(2,5,6,8)(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24)(25,27,29,33)(26,30,32,36)(28,34,35,39)(31,37,38,40)(41,44)(42,43)(45,52)(46,49)(47,51)(48,50)(53,54)(55,56);(9,12)(10,16)(11,18)(13,19)(14,21)(15,22)(17,23)(20,24)(25,28,29,35)(26,31,32,38)(27,34,33,39)(30,37,36,40)(41,45,46,55)(42,48,47,53)(43,50,51,54)(44,52,49,56)Polycyclic presentation:
2^9|a^2=fgh,b^2=fi,c^2=h,d^2=ghi,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4604415);