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 | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1353 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(9,10,14,12)(11,17,19,15)(13,20,21,16)(18,22,24,23)(25,26)(27,35)(28,39)(29,31)(30,44)(32,38)(33,41)(34,43)(36,51)(37,49)(40,50)(42,48)(45,54)(46,56)(47,53)(52,55);(1,2)(3,7)(4,6)(5,8)(9,11,14,19)(10,15,12,17)(13,18,21,24)(16,22,20,23)(26,31)(32,45)(33,39)(34,44)(35,49)(46,47)(48,55)(50,51);(1,3)(2,5)(4,8)(6,7)(9,12,14,10)(11,17,19,15)(13,16,21,20)(18,22,24,23)(25,27,30,38,29,37,43,54)(26,32,34,35,31,45,44,49)(28,40,42,56,41,36,52,53)(33,47,48,51,39,46,55,50);(3,8)(5,7)(9,13,14,21)(10,16,12,20)(11,18,19,24)(15,22,17,23)(25,28,29,41)(26,33,31,39)(27,36,37,40)(30,42,43,52)(32,46,45,47)(34,48,44,55)(35,50,49,51)(38,53,54,56);(2,6)(5,7)(11,19)(15,17)(18,24)(22,23)(28,41)(33,39)(36,40)(42,52)(46,47)(48,55)(50,51)(53,56)Polycyclic presentation:
2^9|a^2=fi,b^2=fi,c^2=fgi,d^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,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, 6818913);