Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682532 |
| 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 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1331 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)(3,5)(4,6)(7,8)(9,10)(11,17)(12,16)(13,15)(14,19)(18,24)(20,22)(21,23)(25,26,30,35)(27,37,40,32)(28,41,43,48)(29,36,31,34)(33,42,49,44)(38,52,54,55)(39,47,45,50)(46,53,51,56);(1,3)(2,5)(4,7)(6,8)(9,11,13,19)(10,14,15,17)(12,20,21,18)(16,24,23,22)(25,27,30,40)(26,32,35,37)(28,38,43,54)(29,45,31,39)(33,46,49,51)(34,50,36,47)(41,55,48,52)(42,56,44,53);(2,6)(3,7)(25,28,29,42,30,43,31,44)(26,33,34,48,35,49,36,41)(27,38,45,56,40,54,39,53)(32,46,50,52,37,51,47,55);(10,15)(14,17)(16,23)(22,24)(25,29,30,31)(26,34,35,36)(27,39,40,45)(28,42,43,44)(32,47,37,50)(33,48,49,41)(38,53,54,56)(46,55,51,52);(3,7)(5,8)(9,12,13,21)(10,16,15,23)(11,18,19,20)(14,22,17,24)(27,40)(28,43)(32,37)(33,49)(39,45)(41,48)(42,44)(47,50)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=ghi,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,e^c=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6672151);