Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5956507 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
1276 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)(11,17)(12,19)(14,20)(15,22)(25,26)(27,31)(28,36)(29,30)(32,37)(33,35)(34,39)(38,40)(41,42,45,48)(43,49,50,46)(44,47,52,54)(51,55,56,53);(1,2,4,6)(3,8,7,5)(11,17)(14,20)(18,23)(21,24)(26,30)(28,37)(31,33)(34,40)(43,50)(46,49)(51,56)(53,55);(1,2)(3,5)(4,6)(7,8)(9,10,13,16)(11,14,17,20)(12,15,19,22)(18,21,23,24)(25,26)(27,33)(28,32)(29,30)(31,35)(34,39)(36,37)(38,40)(41,43)(42,46)(44,51)(45,50)(47,53)(48,49)(52,56)(54,55);(1,3)(2,5)(4,7)(6,8)(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24)(25,27,29,35)(26,31,30,33)(28,34,37,40)(32,38,36,39)(41,42)(43,49)(44,47)(45,48)(46,50)(51,55)(52,54)(53,56);(9,12)(10,15)(11,18)(13,19)(14,21)(16,22)(17,23)(20,24)(25,28,29,37)(26,32,30,36)(27,34,35,40)(31,38,33,39)(41,44,45,52)(42,47,48,54)(43,51,50,56)(46,53,49,55)Polycyclic presentation:
2^9|a^2=g,b^2=fgh,c^2=fhi,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=ef,d^c=dfg62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5926395);