Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470355 |
| 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×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1418 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)(3,7)(4,5)(6,8)(9,10)(11,14)(12,19)(13,15)(16,20)(17,21)(18,24)(22,23)(25,26)(27,36)(28,40)(29,34)(30,46)(31,35)(32,38)(33,43)(37,54)(39,48)(41,53)(42,52)(44,49)(45,51)(47,55)(50,56);(2,5)(6,7)(9,11,13,17)(10,14,15,21)(12,18,20,23)(16,22,19,24)(25,27,29,44,31,39,30,38)(26,32,46,48,35,49,34,36)(28,41,45,56,43,37,42,55)(33,50,51,53,40,47,52,54);(1,3,4,8)(2,6,5,7)(10,15)(11,17)(16,19)(18,23)(25,28)(26,33)(27,37)(29,45)(30,42)(31,43)(32,47)(34,52)(35,40)(36,53)(38,56)(39,41)(44,55)(46,51)(48,54)(49,50);(11,17)(14,21)(18,23)(22,24)(25,29,31,30)(26,34,35,46)(27,38,39,44)(28,42,43,45)(32,48,49,36)(33,51,40,52)(37,55,41,56)(47,53,50,54);(9,12,13,20)(10,16,15,19)(11,18,17,23)(14,22,21,24)(26,35)(32,49)(33,40)(34,46)(36,48)(47,50)(51,52)(53,54)Polycyclic presentation:
2^9|b^2=fg,c^2=gh,d^2=i,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,d^c=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6406593);