Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470185 |
| 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 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1333 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,5,6)(3,7,4,8)(9,10,13,16)(11,14,17,21)(12,19,20,15)(18,24,23,22)(25,26,30,34)(27,31,36,45)(28,33,40,50)(29,32,42,35)(37,44,39,48)(38,46,51,43)(41,47,54,49)(52,56,53,55);(2,6)(7,8)(9,11)(10,14)(12,18)(13,17)(15,22)(16,21)(19,24)(20,23)(25,27)(26,31)(28,39)(29,38)(30,36)(32,46)(33,48)(34,45)(35,43)(37,40)(41,53)(42,51)(44,50)(47,55)(49,56)(52,54);(1,3,5,4)(2,7,6,8)(11,17)(14,21)(18,23)(22,24)(26,32)(27,36)(29,42)(31,43)(33,47)(34,35)(37,39)(41,54)(44,55)(45,46)(48,56)(49,50);(1,4,5,3)(2,8,6,7)(11,17)(14,21)(18,23)(22,24)(25,28,30,40)(26,33,34,50)(27,37,36,39)(29,41,42,54)(31,44,45,48)(32,47,35,49)(38,52,51,53)(43,55,46,56);(9,12,13,20)(10,15,16,19)(11,18,17,23)(14,22,21,24)(26,34)(27,36)(28,40)(32,35)(38,51)(41,54)(44,48)(55,56)Polycyclic presentation:
2^9|a^2=fh,c^2=f,d^2=fi,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,e^b=ei,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6404426);