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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1322 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(9,10,14,17)(11,18,21,15)(12,16,22,13)(19,23,24,20)(25,26)(27,36)(28,33)(29,34)(30,44)(31,35)(32,38)(37,51)(39,46)(40,48)(41,50)(42,47)(43,49)(45,53)(52,55)(54,56);(1,2)(3,7)(4,6)(5,8)(9,11,14,21)(10,15,17,18)(12,19,22,24)(13,23,16,20);(1,3)(2,5)(4,8)(6,7)(9,12)(10,16)(11,19)(13,17)(14,22)(15,20)(18,23)(21,24)(25,27,30,38,31,39,29,42)(26,32,34,36,35,47,44,46)(28,37,40,53,41,54,43,52)(33,45,49,51,50,55,48,56);(3,8)(5,7)(10,17)(12,22)(15,18)(19,24)(25,28,31,41)(26,33,35,50)(27,37,39,54)(29,43,30,40)(32,45,47,55)(34,49,44,48)(36,51,46,56)(38,53,42,52);(2,6)(5,7)(9,13,14,16)(10,12,17,22)(11,20,21,23)(15,24,18,19)(25,29,31,30)(26,34,35,44)(27,38,39,42)(28,40,41,43)(32,46,47,36)(33,48,50,49)(37,52,54,53)(45,51,55,56)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=g,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=dfh,e^c=ei,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6820616);