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 | #58114 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4085 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)(4,8)(5,6)(9,10,13,16)(11,14,17,21)(12,19,20,15)(18,24,23,22)(26,31)(28,39)(29,41)(32,35)(36,49)(37,51)(46,53)(54,55);(1,2,5,6)(3,7,8,4)(11,17)(14,21)(18,23)(22,24)(26,31)(29,41)(32,35)(33,42)(37,51)(40,52)(44,47)(50,56);(1,3)(2,4)(5,8)(6,7)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,20)(23,24)(25,26)(27,35)(28,33)(29,34)(30,31)(32,38)(36,47)(37,48)(39,42)(40,46)(41,43)(44,49)(45,51)(50,55)(52,53)(54,56);(1,4)(2,3)(5,7)(6,8)(25,27,30,38)(26,32,31,35)(28,36,39,49)(29,37,41,51)(33,44,42,47)(34,45,43,48)(40,50,52,56)(46,54,53,55);(1,2)(3,4)(5,6)(7,8)(9,11)(10,14)(12,18)(13,17)(15,22)(16,21)(19,24)(20,23)(25,28,30,39)(26,33,31,42)(27,36,38,49)(29,40,41,52)(32,44,35,47)(34,46,43,53)(37,50,51,56)(45,54,48,55);(1,4)(2,3)(5,7)(6,8)(9,12,13,20)(10,15,16,19)(11,18,17,23)(14,22,21,24)(25,29,30,41)(26,34,31,43)(27,37,38,51)(28,40,39,52)(32,45,35,48)(33,46,42,53)(36,50,49,56)(44,54,47,55)Polycyclic presentation:
2^9|a^2=hi,b^2=g,d^2=gi,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8664787);