Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533123 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1318 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,5,8)(3,7,4,6)(9,10)(11,17)(12,16)(13,15)(14,18)(19,23)(20,22)(21,24)(25,26,29,34,30,33,44,48)(27,35,38,46,37,49,42,31)(28,39,40,47,41,50,43,32)(36,45,53,56,52,55,54,51);(1,3,5,4)(2,6,8,7)(9,11)(10,14)(12,19)(13,18)(15,17)(16,21)(20,24)(22,23)(25,27,30,37)(26,31,33,46)(28,36,41,52)(29,42,44,38)(32,45,47,55)(34,49,48,35)(39,51,50,56)(40,54,43,53);(1,3,5,4)(2,7,8,6)(10,15)(11,18)(16,22)(19,24)(25,28,30,41)(26,32,33,47)(27,36,37,52)(29,43,44,40)(31,45,46,55)(34,50,48,39)(35,51,49,56)(38,54,42,53);(1,4,5,3)(2,6,8,7)(9,12)(10,16)(11,19)(13,20)(14,21)(15,22)(17,23)(18,24)(26,33)(31,46)(32,47)(34,48)(35,49)(39,50)(45,55)(51,56);(12,20)(16,22)(19,24)(21,23)(27,37)(31,46)(35,49)(36,52)(38,42)(45,55)(51,56)(53,54)Polycyclic presentation:
2^9|a^2=fh,b^2=fg,c^2=fg,d^2=fgi,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ei,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=eh71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6478167);