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 | |
3991 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 56 points:
(3,4)(6,7)(10,14)(11,17)(12,20)(22,24)(25,26,31,32)(27,29,34,43)(28,38,42,33)(30,45,46,35)(36,47,39,40)(37,51,49,44)(41,48,55,52)(50,54,53,56);(1,2,5,8)(3,6,4,7)(9,10,13,14)(11,18,17,15)(12,16,20,21)(19,24,23,22)(26,32)(29,43)(33,38)(35,45)(40,47)(44,51)(48,52)(54,56);(1,3,5,4)(2,6,8,7)(9,10)(11,15)(12,16)(13,14)(17,18)(19,22)(20,21)(23,24)(25,27,31,34)(26,29,32,43)(28,39,42,36)(30,37,46,49)(33,47,38,40)(35,44,45,51)(41,53,55,50)(48,56,52,54);(1,2)(3,6)(4,7)(5,8)(9,11,13,17)(10,15,14,18)(12,19,20,23)(16,22,21,24)(25,28,31,42)(26,33,32,38)(27,36,34,39)(29,40,43,47)(30,41,46,55)(35,48,45,52)(37,50,49,53)(44,54,51,56);(1,4,5,3)(2,7,8,6)(9,12,13,20)(10,16,14,21)(11,19,17,23)(15,22,18,24)(25,29)(26,34)(27,32)(28,40)(30,44)(31,43)(33,39)(35,49)(36,38)(37,45)(41,54)(42,47)(46,51)(48,53)(50,52)(55,56);(25,30,31,46)(26,35,32,45)(27,37,34,49)(28,41,42,55)(29,44,43,51)(33,48,38,52)(36,50,39,53)(40,54,47,56)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=gh,d^2=g,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, 8022674);