Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996355 |
| 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 | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1239 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,5,4)(3,7,8,6)(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26,31,36)(27,37,38,48)(28,33,44,51)(29,35,30,34)(32,40,46,41)(39,53,42,56)(43,52,45,50)(47,55,49,54);(1,3,5,8)(2,6,4,7)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(27,38)(32,46)(37,48)(39,42)(40,41)(47,49)(53,56)(54,55);(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(25,27,29,40,31,38,30,41)(26,32,34,48,36,46,35,37)(28,42,45,54,44,39,43,55)(33,49,52,53,51,47,50,56);(1,2,5,4)(3,7,8,6)(12,20)(15,24)(17,19)(21,23)(25,28,31,44)(26,33,36,51)(27,39,38,42)(29,45,30,43)(32,47,46,49)(34,52,35,50)(37,53,48,56)(40,55,41,54);(1,4,5,2)(3,7,8,6)(10,16)(11,18)(15,24)(17,19)(25,29,31,30)(26,34,36,35)(27,40,38,41)(28,43,44,45)(32,48,46,37)(33,50,51,52)(39,54,42,55)(47,53,49,56)Polycyclic presentation:
2^9|a^2=fh,b^2=f,c^2=gi,d^2=fi,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dh,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6958686);