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 | #54414 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4132 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 56 points:
(1,2)(3,4)(5,6)(7,8)(9,10,13,16)(11,14,17,21)(12,15,20,24)(18,22,19,23)(26,31)(27,35)(29,41)(33,43)(37,49)(39,42)(46,52)(55,56);(2,6)(3,7)(11,17)(14,21)(18,19)(22,23)(26,31)(32,36)(33,43)(34,44)(45,51)(46,52)(47,48)(55,56);(1,3,5,7)(2,4,6,8)(9,11)(10,14)(12,18)(13,17)(15,22)(16,21)(19,20)(23,24)(25,26)(27,36)(28,33)(29,34)(30,31)(32,35)(37,51)(38,52)(39,47)(40,43)(41,44)(42,48)(45,49)(46,50)(53,56)(54,55);(1,4)(2,7)(3,6)(5,8)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,23)(20,24)(25,27)(26,32)(28,37)(29,38)(30,35)(31,36)(33,45)(34,46)(39,53)(40,49)(41,50)(42,54)(43,51)(44,52)(47,55)(48,56);(9,11)(10,14)(12,19)(13,17)(15,23)(16,21)(18,20)(22,24)(25,28,30,40)(26,33,31,43)(27,37,35,49)(29,42,41,39)(32,45,36,51)(34,48,44,47)(38,54,50,53)(46,56,52,55);(9,12)(10,15)(11,18)(13,20)(14,22)(16,24)(17,19)(21,23)(25,29,30,41)(26,34,31,44)(27,38,35,50)(28,39,40,42)(32,46,36,52)(33,47,43,48)(37,53,49,54)(45,55,51,56)Polycyclic presentation:
2^9|a^2=ghi,c^2=g,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9820102);