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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1332 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)(27,36)(28,33)(29,34)(30,44)(31,35)(32,40)(37,50)(38,51)(39,45)(41,47)(42,49)(43,48)(46,53)(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)(27,37)(32,45)(36,50)(38,52)(39,40)(46,55)(51,56)(53,54);(1,3)(2,5)(4,8)(6,7)(25,27,31,37)(26,32,35,45)(28,38,42,52)(29,40,30,39)(33,46,49,55)(34,50,44,36)(41,54,43,53)(47,51,48,56);(3,8)(5,7)(9,12,13,21)(10,15,16,24)(11,18,19,20)(14,22,17,23)(25,28,31,42)(26,33,35,49)(27,38,37,52)(29,43,30,41)(32,46,45,55)(34,48,44,47)(36,51,50,56)(39,54,40,53);(2,6)(5,7)(10,16)(14,17)(15,24)(22,23)(25,29,31,30)(26,34,35,44)(27,39,37,40)(28,41,42,43)(32,36,45,50)(33,47,49,48)(38,53,52,54)(46,56,55,51)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=i,d^2=fi,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=ehi,d^c=dhi,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, 7144326);