Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500120 |
| 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 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1283 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2)(3,6)(4,8)(5,7)(25,26,31,44,30,34,29,35)(27,36,43,32,38,51,39,47)(28,33,45,56,41,49,42,50)(37,52,55,46,40,54,53,48);(1,3,5,4)(2,6,7,8)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,27)(26,32)(28,40)(29,43)(30,38)(31,39)(33,48)(34,47)(35,51)(36,44)(37,41)(42,53)(45,55)(46,49)(50,52)(54,56);(2,7)(6,8)(9,11,13,19)(10,14,16,17)(12,20,21,18)(15,23,24,22)(25,28,30,41)(26,33,34,49)(27,37,38,40)(29,42,31,45)(32,46,47,48)(35,50,44,56)(36,52,51,54)(39,53,43,55);(9,12,13,21)(10,15,16,24)(11,18,19,20)(14,22,17,23)(26,34)(28,41)(32,47)(35,44)(36,51)(37,40)(42,45)(53,55);(1,4,5,3)(2,8,7,6)(10,16)(14,17)(15,24)(22,23)(27,38)(28,41)(32,47)(33,49)(36,51)(39,43)(42,45)(50,56)Polycyclic presentation:
2^9|a^2=fi,b^2=g,c^2=h,d^2=hi,e^2=g,f^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=eh,f^b=fi,d^c=dh,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6439601);