Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #392 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4034 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 56 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,14)(12,17)(13,18)(15,19)(16,21)(20,22)(23,24)(25,26,30,31)(27,35,38,32)(28,39,40,33)(29,42,43,34)(36,44,49,47)(37,45,51,48)(41,46,53,52)(50,55,56,54);(1,2,4,6)(3,5,8,7)(10,11)(12,19)(16,18)(20,24)(26,31)(28,40)(32,35)(34,42)(36,49)(41,53)(45,48)(50,56);(1,2)(3,5)(4,6)(7,8)(9,11,14,10)(12,17,19,15)(13,18,21,16)(20,23,24,22)(25,27,30,38)(26,32,31,35)(28,36,40,49)(29,37,43,51)(33,44,39,47)(34,45,42,48)(41,50,53,56)(46,54,52,55);(1,3,4,8)(2,5,6,7)(9,12)(10,15)(11,17)(13,20)(14,19)(16,22)(18,23)(21,24)(25,26)(27,35)(28,33)(29,34)(30,31)(32,38)(36,47)(37,48)(39,40)(41,46)(42,43)(44,49)(45,51)(50,55)(52,53)(54,56);(9,13,14,21)(10,16,11,18)(12,20,19,24)(15,22,17,23)(25,28,30,40)(26,33,31,39)(27,36,38,49)(29,41,43,53)(32,44,35,47)(34,46,42,52)(37,50,51,56)(45,54,48,55);(25,29,30,43)(26,34,31,42)(27,37,38,51)(28,41,40,53)(32,45,35,48)(33,46,39,52)(36,50,49,56)(44,54,47,55)Polycyclic presentation:
2^9|a^2=gi,b^2=gh,c^2=h,d^2=gh,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8022720);