Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| 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 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2402 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1196 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)(3,6)(4,5)(7,8)(9,10,13,15)(11,17,19,14)(12,16,21,23)(18,24,20,22)(25,26)(27,32)(28,33)(29,34)(30,36)(31,35)(37,45)(38,46)(39,51)(40,47)(41,48)(42,49)(43,50)(44,52)(53,55)(54,56);(9,11,13,19)(10,14,15,17)(12,20,21,18)(16,24,23,22)(25,27,31,40)(26,32,35,47)(28,41,43,53)(29,38,30,39)(33,48,50,55)(34,46,36,51)(37,44,54,42)(45,52,56,49);(2,5)(6,8)(25,28,31,43)(26,33,35,50)(27,37,40,54)(29,42,30,44)(32,45,47,56)(34,49,36,52)(38,53,39,41)(46,55,51,48);(1,3,4,7)(2,6,5,8)(10,15)(14,17)(16,23)(22,24)(25,29,31,30)(26,34,35,36)(27,38,40,39)(28,42,43,44)(32,46,47,51)(33,49,50,52)(37,53,54,41)(45,55,56,48);(9,12,13,21)(10,16,15,23)(11,18,19,20)(14,22,17,24)(26,35)(27,39)(29,30)(32,46)(33,50)(37,41)(38,40)(42,44)(45,55)(47,51)(48,56)(53,54)Polycyclic presentation:
2^9|a^2=f,b^2=fi,c^2=i,d^2=gi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6611854);