Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682440 |
| 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 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1191 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)(11,17)(12,16)(13,15)(14,19)(18,24)(20,22)(21,23)(25,26,31,35)(27,36,40,53)(28,33,41,48)(29,34,30,45)(32,38,47,39)(37,52,42,56)(43,50,44,51)(46,55,49,54);(1,2)(3,6)(4,5)(7,8)(9,11,13,19)(10,14,15,17)(12,20,21,18)(16,24,23,22)(28,41)(33,48)(37,42)(43,44)(46,49)(50,51)(52,56)(54,55);(2,5)(6,8)(25,27,30,39,31,40,29,38)(26,32,34,36,35,47,45,53)(28,42,43,55,41,37,44,54)(33,49,51,56,48,46,50,52);(1,3,4,7)(2,6,5,8)(10,15)(14,17)(16,23)(22,24)(25,28)(26,33)(27,37)(29,44)(30,43)(31,41)(32,46)(34,51)(35,48)(36,52)(38,55)(39,54)(40,42)(45,50)(47,49)(53,56);(2,5)(6,8)(9,12,13,21)(10,16,15,23)(11,18,19,20)(14,22,17,24)(25,29,31,30)(26,34,35,45)(27,38,40,39)(28,43,41,44)(32,36,47,53)(33,50,48,51)(37,54,42,55)(46,56,49,52)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=g,d^2=h,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=di,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6695153);