Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7234485 |
| 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 | |
1356 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(9,10,13,16)(11,17,19,14)(12,15,21,24)(18,23,20,22)(25,26)(27,37)(28,34)(29,36)(30,32)(31,35)(33,41)(38,51)(39,45)(40,48)(42,46)(43,50)(44,49)(47,53)(52,56)(54,55);(1,2)(3,7)(4,6)(5,8)(9,11,13,19)(10,14,16,17)(12,20,21,18)(15,23,24,22)(26,32)(28,42)(33,45)(35,36)(37,48)(38,52)(43,44)(53,54);(1,3)(2,5)(4,8)(6,7)(25,27,30,40)(26,33,32,45)(28,38,42,52)(29,41,31,39)(34,47,46,55)(35,37,36,48)(43,54,44,53)(49,56,50,51);(3,8)(5,7)(9,12,13,21)(10,15,16,24)(11,18,19,20)(14,22,17,23)(25,28,30,42)(26,34,32,46)(27,38,40,52)(29,44,31,43)(33,47,45,55)(35,50,36,49)(37,51,48,56)(39,54,41,53);(2,6)(5,7)(10,16)(14,17)(15,24)(22,23)(25,29,30,31)(26,35,32,36)(27,39,40,41)(28,43,42,44)(33,48,45,37)(34,49,46,50)(38,53,52,54)(47,51,55,56)Polycyclic presentation:
2^9|a^2=fi,b^2=fi,c^2=i,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^c=ei,g^c=gi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7144254);