Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996719 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1174 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2)(3,6)(4,5)(7,8)(12,20)(15,21)(19,24)(22,23)(25,26,31,36)(27,32,30,35)(28,39,42,46)(29,34,44,49)(33,37,48,40)(38,45,43,50)(41,52,54,55)(47,51,56,53);(2,5)(3,7)(9,10,13,16)(11,17,18,14)(12,21,20,15)(19,22,24,23)(25,27,31,30)(26,32,36,35)(28,40,42,37)(29,38,44,43)(33,46,48,39)(34,45,49,50)(41,53,54,51)(47,55,56,52);(9,11)(10,14)(12,19)(13,18)(15,22)(16,17)(20,24)(21,23)(25,28,27,37,31,42,30,40)(26,33,35,46,36,48,32,39)(29,41,38,51,44,54,43,53)(34,47,50,55,49,56,45,52);(1,3,4,7)(2,6,5,8)(11,18)(14,17)(19,24)(22,23)(25,29)(26,34)(27,38)(28,41)(30,43)(31,44)(32,45)(33,47)(35,50)(36,49)(37,51)(39,52)(40,53)(42,54)(46,55)(48,56);(9,12,13,20)(10,15,16,21)(11,19,18,24)(14,22,17,23)(25,27,31,30)(26,35,36,32)(28,37,42,40)(29,43,44,38)(33,46,48,39)(34,45,49,50)(41,53,54,51)(47,52,56,55)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=gi,d^2=f,e^2=h,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^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7093612);