Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6433044 |
| 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 | #2619 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1406 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,16)(13,15)(17,20)(18,21)(19,22)(23,24)(25,26)(27,37)(28,42)(29,34)(30,35)(31,36)(32,41)(33,44)(38,46)(39,47)(40,48)(43,50)(45,49)(51,52)(53,56)(54,55);(2,5)(6,8)(9,11)(10,14)(12,18)(13,17)(15,20)(16,21)(19,23)(22,24)(25,27,30,39,31,40,29,41)(26,32,34,48,36,47,35,37)(28,38,44,52,45,53,43,54)(33,46,42,55,50,56,49,51);(1,3,4,7)(2,6,5,8)(11,17)(14,20)(18,23)(21,24)(25,28,29,43,31,45,30,44)(26,33,35,49,36,50,34,42)(27,38,41,54,40,53,39,52)(32,46,37,51,47,56,48,55);(10,15)(14,20)(16,22)(21,24)(25,29,31,30)(26,34,36,35)(27,39,40,41)(28,43,45,44)(32,37,47,48)(33,42,50,49)(38,52,53,54)(46,51,56,55);(9,12,13,19)(10,16,15,22)(11,18,17,23)(14,21,20,24)(27,40)(32,47)(37,48)(38,53)(39,41)(46,56)(51,55)(52,54)Polycyclic presentation:
2^9|b^2=g,c^2=f,d^2=i,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6344362);