Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470137 |
| 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 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1326 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)(12,15)(13,16)(17,21)(18,22)(19,23)(20,24)(25,26,30,36,31,34,29,35)(27,37,41,32,39,49,40,48)(28,42,45,51,46,52,44,33)(38,50,55,56,43,47,54,53);(2,5)(6,8)(9,11)(10,14)(12,19)(13,17)(15,23)(16,21)(18,20)(22,24)(25,27)(26,32)(28,43)(29,41)(30,40)(31,39)(33,50)(34,48)(35,49)(36,37)(38,46)(42,53)(44,54)(45,55)(47,51)(52,56);(11,17)(14,21)(18,19)(22,23)(25,28,31,46)(26,33,34,51)(27,38,39,43)(29,45,30,44)(32,47,48,50)(35,42,36,52)(37,53,49,56)(40,55,41,54);(9,12,13,20)(10,15,16,24)(11,18,17,19)(14,22,21,23)(27,39)(32,48)(37,49)(38,43)(40,41)(47,50)(53,56)(54,55);(1,3,4,7)(2,6,5,8)(10,16)(14,21)(15,24)(22,23)(26,34)(27,39)(33,51)(35,36)(38,43)(40,41)(42,52)(54,55)Polycyclic presentation:
2^9|a^2=g,c^2=i,d^2=hi,e^2=fg,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh,e^b=ei,f^b=fi,g^b=gi,f^c=fi,g^c=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6393393);