Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996375 |
| 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×D8 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2402 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1226 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)(3,6,4,7)(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26,29,35,30,34,41,31)(27,36,39,43,38,32,45,47)(28,33,40,49,42,48,53,44)(37,50,51,54,52,46,55,56);(1,3,5,4)(2,6,8,7)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(26,31)(28,40)(29,41)(32,43)(34,35)(36,47)(37,51)(39,45)(42,53)(44,49)(52,55)(54,56);(1,3,5,4)(2,7,8,6)(9,12,13,20)(10,15,16,24)(11,17,18,19)(14,21,22,23)(25,27)(26,32)(28,37)(29,39)(30,38)(31,43)(33,46)(34,36)(35,47)(40,51)(41,45)(42,52)(44,54)(48,50)(49,56)(53,55);(1,3,5,4)(2,6,8,7)(12,20)(15,24)(17,19)(21,23)(25,28,41,53,30,42,29,40)(26,33,31,44,34,48,35,49)(27,37,45,55,38,52,39,51)(32,46,43,54,36,50,47,56);(1,4,5,3)(2,7,8,6)(10,16)(11,18)(15,24)(17,19)(26,34)(27,38)(31,35)(33,48)(37,52)(39,45)(44,49)(51,55)Polycyclic presentation:
2^9|a^2=fghi,b^2=gi,c^2=ghi,d^2=fg,e^2=gi,f^2=i,b^a=bf,c^a=cg,e^a=ehi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7031004);