Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470428 |
| 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 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1299 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)(3,8)(4,6)(5,7)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,23)(20,24)(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);(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(25,27,30,41,31,38,29,40)(26,32,35,37,36,46,34,48)(28,42,43,55,44,39,45,54)(33,49,50,56,51,47,52,53);(1,3,5,4)(2,6,7,8)(9,12,13,20)(10,15,16,24)(11,17,18,19)(14,21,22,23)(27,38)(32,46)(37,48)(39,42)(40,41)(47,49)(53,56)(54,55);(1,4,5,3)(2,6,7,8)(11,18)(14,22)(17,19)(21,23)(25,28)(26,33)(27,39)(29,45)(30,43)(31,44)(32,47)(34,52)(35,50)(36,51)(37,53)(38,42)(40,55)(41,54)(46,49)(48,56);(10,16)(14,22)(15,24)(21,23)(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=i,b^2=f,c^2=ghi,d^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6387375);