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 | |
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)(25,26,30,34)(27,35,36,52)(28,32,43,51)(29,44,37,33)(31,45,46,39)(38,53,40,56)(41,50,42,49)(47,54,48,55);(1,2,5,6)(3,7,4,8)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(27,36)(31,46)(35,52)(38,40)(39,45)(47,48)(53,56)(54,55);(1,3,5,4)(2,7,6,8)(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(25,27,37,45,30,36,29,39)(26,31,44,52,34,46,33,35)(28,40,41,55,43,38,42,54)(32,48,49,53,51,47,50,56);(1,3,5,4)(2,8,6,7)(12,20)(15,24)(17,19)(21,23)(25,28,30,43)(26,32,34,51)(27,38,36,40)(29,42,37,41)(31,47,46,48)(33,50,44,49)(35,53,52,56)(39,55,45,54);(1,4,5,3)(2,8,6,7)(10,16)(11,18)(15,24)(17,19)(25,29,30,37)(26,33,34,44)(27,39,36,45)(28,41,43,42)(31,35,46,52)(32,49,51,50)(38,54,40,55)(47,56,48,53)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=fgi,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, 6960222);