Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907993 |
| 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 | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2285 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1367 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(9,10)(11,17)(12,16)(13,15)(14,19)(18,24)(20,22)(21,23)(25,26,31,35)(27,36,37,52)(28,33,41,48)(29,34,30,45)(32,39,46,40)(38,53,42,56)(43,50,44,51)(47,55,49,54);(1,2)(3,7)(4,6)(5,8)(9,11,13,19)(10,14,15,17)(12,20,21,18)(16,24,23,22)(27,37)(28,41)(32,46)(33,48)(36,52)(39,40)(43,44)(50,51);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,16)(13,15)(17,19)(18,22)(20,24)(21,23)(25,27,29,39,31,37,30,40)(26,32,45,52,35,46,34,36)(28,42,44,54,41,38,43,55)(33,49,50,53,48,47,51,56);(3,8)(5,7)(10,15)(14,17)(16,23)(22,24)(25,28,31,41)(26,33,35,48)(27,38,37,42)(29,44,30,43)(32,47,46,49)(34,51,45,50)(36,53,52,56)(39,55,40,54);(2,6)(5,7)(9,12,13,21)(10,16,15,23)(11,18,19,20)(14,22,17,24)(25,29,31,30)(26,34,35,45)(27,39,37,40)(28,43,41,44)(32,36,46,52)(33,50,48,51)(38,54,42,55)(47,56,49,53)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=gi,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=efhi,d^c=dfh,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6820444);