Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500434 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2280 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1491 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2)(3,6)(4,5)(7,8)(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26)(27,37)(28,42)(29,34)(30,35)(31,36)(32,39)(33,43)(38,46)(40,48)(41,47)(44,50)(45,49)(51,52)(53,55)(54,56);(2,5)(6,8)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(25,27,30,40,31,41,29,39)(26,32,34,47,36,48,35,37)(28,38,43,52,45,54,44,53)(33,46,42,55,50,56,49,51);(1,3,4,7)(2,6,5,8)(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(25,28,29,44,31,45,30,43)(26,33,35,49,36,50,34,42)(27,38,39,53,41,54,40,52)(32,46,37,51,48,56,47,55);(12,20)(15,24)(17,19)(21,23)(25,29,31,30)(26,34,36,35)(27,39,41,40)(28,43,45,44)(32,47,48,37)(33,49,50,42)(38,52,54,53)(46,51,56,55);(11,18)(14,22)(17,19)(21,23)(25,29,31,30)(26,34,36,35)(27,39,41,40)(28,44,45,43)(32,47,48,37)(33,42,50,49)(38,53,54,52)(46,55,56,51)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=f,d^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=eh,d^c=dhi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6417919);