Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907869 |
| 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 | |
1337 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)(12,20,21,15)(18,22,24,23)(25,26)(27,36)(28,40)(29,35)(30,34)(31,45)(32,38)(33,43)(37,51)(39,48)(41,55)(42,49)(44,47)(46,56)(50,52)(53,54);(1,2)(3,7)(4,6)(5,8)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24)(25,27)(26,32)(28,37)(29,44)(30,38)(31,39)(33,46)(34,36)(35,48)(40,54)(41,53)(42,56)(43,52)(45,47)(49,50)(51,55);(1,3)(2,5)(4,8)(6,7)(12,21)(15,20)(18,24)(22,23)(25,28,31,43,30,41,29,42)(26,33,35,40,34,49,45,55)(27,37,39,52,38,53,44,56)(32,46,48,54,36,50,47,51);(3,8)(5,7)(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(26,34)(27,38)(28,41)(35,45)(39,44)(42,43)(46,50)(51,54);(2,6)(5,7)(11,19)(14,17)(18,24)(22,23)(25,29,30,31)(26,35,34,45)(27,39,38,44)(28,42,41,43)(32,47,36,48)(33,40,49,55)(37,52,53,56)(46,51,50,54)Polycyclic presentation:
2^9|a^2=fi,c^2=g,d^2=fi,e^2=i,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=dfh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6817040);