Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907993 |
| 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 | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1355 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(9,10,15,17)(11,18,21,16)(12,13,22,14)(19,23,24,20)(25,26)(27,36)(28,33)(29,34)(30,45)(31,35)(32,40)(37,52)(38,53)(39,46)(41,56)(42,49)(43,51)(44,50)(47,54)(48,55);(1,2)(3,7)(4,6)(5,8)(9,11,15,21)(10,16,17,18)(12,19,22,24)(13,20,14,23)(27,37)(32,46)(36,52)(38,41)(39,40)(47,48)(53,56)(54,55);(1,3)(2,5)(4,8)(6,7)(9,12)(10,13)(11,19)(14,17)(15,22)(16,20)(18,23)(21,24)(25,27,29,39,31,37,30,40)(26,32,45,52,35,46,34,36)(28,41,44,54,43,38,42,55)(33,48,49,53,51,47,50,56);(3,8)(5,7)(9,13,15,14)(10,12,17,22)(11,20,21,23)(16,19,18,24)(25,28,31,43)(26,33,35,51)(27,38,37,41)(29,44,30,42)(32,47,46,48)(34,50,45,49)(36,53,52,56)(39,55,40,54);(2,6)(5,7)(9,14,15,13)(10,12,17,22)(11,20,21,23)(16,24,18,19)(25,29,31,30)(26,34,35,45)(27,39,37,40)(28,42,43,44)(32,36,46,52)(33,49,51,50)(38,54,41,55)(47,56,48,53)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=gi,d^2=fi,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,d^c=dfh,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6822092);