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 | |
4022 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)(9,10,15,16)(11,19,20,17)(12,23,24,13)(14,21,22,18)(25,26,31,36)(27,32,37,41)(28,40,43,33)(29,34,44,38)(30,35,47,52)(39,48,46,51)(42,54,55,49)(45,50,56,53);(1,2,4,6)(3,5,8,7)(10,16)(11,20)(13,23)(18,21)(27,37)(29,44)(32,41)(34,38)(39,46)(45,56)(48,51)(50,53);(1,2)(3,5)(4,6)(7,8)(9,11,15,20)(10,17,16,19)(12,21,24,18)(13,22,23,14)(25,27)(26,32)(28,38)(29,40)(30,46)(31,37)(33,44)(34,43)(35,51)(36,41)(39,47)(42,50)(45,49)(48,52)(53,55)(54,56);(1,3)(2,5)(4,8)(6,7)(9,12)(10,13)(11,21)(14,19)(15,24)(16,23)(17,22)(18,20)(25,28,31,43)(26,33,36,40)(27,38,37,34)(29,32,44,41)(30,42,47,55)(35,49,52,54)(39,53,46,50)(45,48,56,51);(9,13)(10,12)(11,22)(14,20)(15,23)(16,24)(17,21)(18,19)(25,29,31,44)(26,34,36,38)(27,33,37,40)(28,41,43,32)(30,45,47,56)(35,50,52,53)(39,49,46,54)(42,51,55,48);(9,14,15,22)(10,18,16,21)(11,13,20,23)(12,19,24,17)(25,30)(26,35)(27,39)(28,42)(29,45)(31,47)(32,48)(33,49)(34,50)(36,52)(37,46)(38,53)(40,54)(41,51)(43,55)(44,56)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=h,d^2=gi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,e^c=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, 8947304);