Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| 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 | |
1262 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)(11,17)(16,21)(18,23)(25,26)(27,30)(28,33)(29,34)(31,35)(32,37)(36,38)(39,40)(41,42,46,49)(43,50,47,44)(45,53,54,48)(51,52,55,56);(1,2,4,6)(3,5,7,8)(11,17)(15,20)(18,23)(22,24)(26,27)(28,35)(29,37)(38,39)(43,47)(44,50)(51,55)(52,56);(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,27,30,26)(28,31,35,33)(29,34,37,32)(36,38,40,39)(41,43,46,47)(42,44,49,50)(45,51,54,55)(48,52,53,56);(1,3)(2,5)(4,7)(6,8)(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24)(25,28,30,35)(26,31,27,33)(29,36,37,40)(32,38,34,39)(41,44)(42,47)(43,49)(45,52)(46,50)(48,55)(51,53)(54,56);(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)(25,29)(26,32)(27,34)(28,36)(30,37)(31,38)(33,39)(35,40)(41,45)(42,48)(43,51)(44,52)(46,54)(47,55)(49,53)(50,56)Polycyclic presentation:
2^9|a^2=fh,b^2=gi,c^2=h,d^2=f,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4389285);