Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7309060 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1155 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26,30,36,31,37,29,35)(27,32,39,48,40,49,44,54)(28,34,42,51,43,53,45,33)(38,47,56,50,41,52,55,46);(2,6)(3,4)(9,11,13,19)(10,14,16,17)(12,20,21,18)(15,23,24,22)(25,27,31,40)(26,32,37,49)(28,41,43,38)(29,44,30,39)(33,50,51,46)(34,52,53,47)(35,54,36,48)(42,55,45,56);(1,2)(3,8)(4,7)(5,6)(25,28,31,43)(26,33,37,51)(27,38,40,41)(29,42,30,45)(32,46,49,50)(34,36,53,35)(39,55,44,56)(47,48,52,54);(1,3,5,4)(2,7,6,8)(9,12,13,21)(10,15,16,24)(11,18,19,20)(14,22,17,23)(25,28)(26,34)(27,38)(29,45)(30,42)(31,43)(32,47)(33,35)(36,51)(37,53)(39,56)(40,41)(44,55)(46,54)(48,50)(49,52);(1,4,5,3)(2,7,6,8)(10,16)(14,17)(15,24)(22,23)(25,29,31,30)(26,35,37,36)(27,39,40,44)(28,42,43,45)(32,48,49,54)(33,34,51,53)(38,55,41,56)(46,52,50,47)Polycyclic presentation:
2^9|a^2=g,b^2=fi,c^2=i,d^2=fhi,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfh,e^b=eh,d^c=dh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7397505);