Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535655 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1265 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,5,6)(3,7,4,8)(9,10)(11,14)(12,16)(13,15)(17,21)(18,20)(19,22)(23,24)(25,26)(27,37)(28,42)(29,34)(30,35)(31,36)(32,41)(33,43)(38,46)(39,48)(40,47)(44,50)(45,49)(51,53)(52,56)(54,55);(2,6)(7,8)(9,11,13,17)(10,14,15,21)(12,18,19,23)(16,20,22,24)(25,27,30,41,31,39,29,40)(26,32,34,37,36,47,35,48)(28,38,44,54,45,52,43,53)(33,46,49,51,50,56,42,55);(1,3,5,4)(2,7,6,8)(10,15)(11,17)(16,22)(18,23)(25,28,29,43,31,45,30,44)(26,33,35,42,36,50,34,49)(27,38,40,53,39,52,41,54)(32,46,48,55,47,56,37,51);(1,4,5,3)(2,8,6,7)(9,12,13,19)(10,16,15,22)(11,18,17,23)(14,20,21,24)(27,39)(32,47)(37,48)(38,52)(40,41)(46,56)(51,55)(53,54);(12,19)(16,22)(18,23)(20,24)(25,29,31,30)(26,34,36,35)(27,40,39,41)(28,43,45,44)(32,37,47,48)(33,49,50,42)(38,53,52,54)(46,51,56,55)Polycyclic presentation:
2^9|a^2=fgh,b^2=gi,c^2=fi,d^2=fg,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^d=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6476188);