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 | #56414 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4089 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)(4,5)(6,8)(10,14)(11,17)(16,21)(18,23)(26,31)(27,35)(28,38)(34,41)(37,49)(40,52)(45,47)(55,56);(2,5)(6,7)(9,10,13,14)(11,15,17,20)(12,16,19,21)(18,22,23,24)(26,31)(28,38)(32,43)(34,41)(36,39)(40,52)(46,51)(50,53);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,26)(27,32)(28,33)(29,41)(30,31)(34,42)(35,43)(36,45)(37,51)(38,44)(39,47)(40,54)(46,49)(48,52)(50,56)(53,55);(1,3)(2,6)(4,8)(5,7)(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,27,30,35)(26,32,31,43)(28,39,38,36)(29,37,42,49)(33,47,44,45)(34,46,41,51)(40,53,52,50)(48,56,54,55);(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24)(25,28,30,38)(26,33,31,44)(27,36,35,39)(29,40,42,52)(32,45,43,47)(34,48,41,54)(37,50,49,53)(46,55,51,56);(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)(25,29,30,42)(26,34,31,41)(27,37,35,49)(28,40,38,52)(32,46,43,51)(33,48,44,54)(36,50,39,53)(45,55,47,56)Polycyclic presentation:
2^9|b^2=ghi,d^2=gi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8324449);