Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| 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 | |
1243 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(2,6)(7,8)(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26,31,36)(27,37,38,49)(28,33,42,48)(29,35,30,34)(32,43,45,40)(39,51,52,56)(41,50,44,47)(46,53,55,54);(1,2,5,6)(3,7,4,8)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(27,38)(32,45)(37,49)(39,52)(40,43)(46,55)(51,56)(53,54);(1,3,5,4)(2,7,6,8)(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(25,27,30,40,31,38,29,43)(26,32,35,37,36,45,34,49)(28,39,41,54,42,52,44,53)(33,46,47,51,48,55,50,56);(1,3,5,4)(2,8,6,7)(12,20)(15,24)(17,19)(21,23)(25,28,31,42)(26,33,36,48)(27,39,38,52)(29,44,30,41)(32,46,45,55)(34,50,35,47)(37,51,49,56)(40,54,43,53);(1,4,5,3)(2,8,6,7)(10,16)(11,18)(15,24)(17,19)(25,29,31,30)(26,34,36,35)(27,40,38,43)(28,41,42,44)(32,37,45,49)(33,47,48,50)(39,53,52,54)(46,56,55,51)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=fg,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=dhi,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6960538);