Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996375 |
| 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 | #2414 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1362 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)(3,7)(4,6)(5,8)(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26,31,36,32,37,30,35)(27,38,40,51,41,50,29,33)(28,42,45,54,46,34,48,53)(39,56,44,49,43,55,47,52);(1,3)(2,5)(4,8)(6,7)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(25,27,32,41)(26,33,37,51)(28,43,46,39)(29,30,40,31)(34,52,42,49)(35,38,36,50)(44,48,47,45)(53,55,54,56);(2,6)(5,7)(9,12,13,20)(10,15,16,24)(11,17,18,19)(14,21,22,23)(25,28)(26,34)(27,39)(29,47)(30,48)(31,45)(32,46)(33,49)(35,54)(36,53)(37,42)(38,55)(40,44)(41,43)(50,56)(51,52);(3,8)(5,7)(12,20)(15,24)(17,19)(21,23)(25,29,31,27,32,40,30,41)(26,33,36,38,37,51,35,50)(28,44,45,43,46,47,48,39)(34,52,53,56,42,49,54,55);(10,16)(11,18)(15,24)(17,19)(25,30,32,31)(26,35,37,36)(27,29,41,40)(28,45,46,48)(33,50,51,38)(34,53,42,54)(39,44,43,47)(49,55,52,56)Polycyclic presentation:
2^9|a^2=fgh,b^2=i,c^2=hi,d^2=fgi,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ehi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7018190);