Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6469956 |
| 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 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1332 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)(9,10,13,15)(11,14,17,21)(12,19,20,16)(18,24,23,22)(25,26)(27,36)(28,33)(29,34)(30,45)(31,35)(32,39)(37,53)(38,47)(40,52)(41,56)(42,49)(43,50)(44,51)(46,54)(48,55);(9,11)(10,14)(12,18)(13,17)(15,21)(16,22)(19,24)(20,23)(25,27,31,40)(26,32,35,47)(28,41,43,37)(29,39,30,38)(33,48,50,46)(34,52,45,36)(42,54,44,55)(49,56,51,53);(1,3,4,8)(2,5,6,7)(10,15)(11,17)(16,19)(18,23)(25,28,31,43)(26,33,35,50)(27,37,40,41)(29,44,30,42)(32,46,47,48)(34,51,45,49)(36,53,52,56)(38,55,39,54);(11,17)(14,21)(18,23)(22,24)(25,29,31,30)(26,34,35,45)(27,38,40,39)(28,42,43,44)(32,36,47,52)(33,49,50,51)(37,54,41,55)(46,56,48,53);(9,12,13,20)(10,16,15,19)(11,18,17,23)(14,22,21,24)(26,35)(32,47)(33,50)(34,45)(36,52)(46,48)(49,51)(53,56)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=gh,d^2=i,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6406422);