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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1308 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,4,6)(3,7,8,5)(12,20)(15,21)(19,24)(22,23)(25,26,31,44,30,35,29,36)(27,33,43,32,38,49,39,47)(28,34,45,55,42,50,41,51)(37,48,54,46,52,56,40,53);(1,3,4,8)(2,5,6,7)(9,10,13,16)(11,17,18,14)(12,21,20,15)(19,22,24,23)(25,27,30,38)(26,32,35,47)(28,40,42,54)(29,43,31,39)(33,36,49,44)(34,48,50,56)(37,45,52,41)(46,55,53,51);(2,6)(5,7)(9,11)(10,14)(12,19)(13,18)(15,22)(16,17)(20,24)(21,23)(25,27)(26,33)(28,37)(29,39)(30,38)(31,43)(32,44)(34,48)(35,49)(36,47)(40,41)(42,52)(45,54)(46,55)(50,56)(51,53);(3,8)(5,7)(11,18)(14,17)(19,24)(22,23)(25,28,29,41,30,42,31,45)(26,34,36,51,35,50,44,55)(27,37,39,40,38,52,43,54)(32,46,33,48,47,53,49,56);(9,12,13,20)(10,15,16,21)(11,19,18,24)(14,22,17,23)(26,35)(27,38)(34,50)(36,44)(37,52)(39,43)(40,54)(51,55)Polycyclic presentation:
2^9|a^2=fi,b^2=gh,d^2=fg,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7096180);