Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470860 |
| 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 | |
1178 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,5,8)(3,7,4,6)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,23)(20,24)(25,26)(27,37)(28,42)(29,36)(30,35)(31,34)(32,40)(33,45)(38,47)(39,49)(41,48)(43,50)(44,51)(46,52)(53,55)(54,56);(1,3,5,4)(2,6,8,7)(9,11)(10,14)(12,19)(13,17)(15,23)(16,21)(18,20)(22,24)(25,27,29,40,31,39,30,41)(26,32,35,37,34,48,36,49)(28,43,44,55,46,38,45,54)(33,50,52,56,51,47,42,53);(1,3,5,4)(2,7,8,6)(11,17)(14,21)(18,19)(22,23)(25,28,30,45,31,46,29,44)(26,33,36,42,34,51,35,52)(27,38,41,55,39,43,40,54)(32,47,49,56,48,50,37,53);(9,12,13,20)(10,15,16,24)(11,18,17,19)(14,22,21,23)(27,39)(32,48)(37,49)(38,43)(40,41)(47,50)(53,56)(54,55);(1,4,5,3)(2,6,8,7)(10,16)(14,21)(15,24)(22,23)(26,34)(27,39)(33,51)(35,36)(38,43)(40,41)(42,52)(54,55)Polycyclic presentation:
2^9|a^2=fg,b^2=f,c^2=fi,d^2=hi,e^2=fg,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^b=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6393676);