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 | #57481 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4102 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 56 points:
(1,2,5,6)(3,4,7,8)(9,10,15,16)(11,13,12,17)(14,18,22,23)(19,21,20,24)(25,26,30,34)(27,35,36,31)(28,39,40,32)(29,33,43,48)(37,44,50,49)(38,47,42,45)(41,52,53,46)(51,55,54,56);(2,6)(3,7)(10,16)(11,12)(18,23)(19,20)(27,36)(28,40)(31,35)(32,39)(38,42)(41,53)(45,47)(46,52);(1,3)(2,4)(5,7)(6,8)(9,11,15,12)(10,17,16,13)(14,19,22,20)(18,24,23,21)(25,27,30,36)(26,31,34,35)(28,37,40,50)(29,42,43,38)(32,44,39,49)(33,47,48,45)(41,54,53,51)(46,56,52,55);(1,4)(2,3)(5,8)(6,7)(9,12,15,11)(10,13,16,17)(14,20,22,19)(18,21,23,24)(25,26)(27,31)(28,39)(29,33)(30,34)(32,40)(35,36)(37,49)(38,45)(41,52)(42,47)(43,48)(44,50)(46,53)(51,56)(54,55);(9,13)(10,11)(12,16)(14,21)(15,17)(18,19)(20,23)(22,24)(25,28,30,40)(26,32,34,39)(27,37,36,50)(29,41,43,53)(31,44,35,49)(33,46,48,52)(38,51,42,54)(45,55,47,56);(9,14,15,22)(10,18,16,23)(11,19,12,20)(13,21,17,24)(25,29,30,43)(26,33,34,48)(27,38,36,42)(28,41,40,53)(31,45,35,47)(32,46,39,52)(37,51,50,54)(44,55,49,56)Polycyclic presentation:
2^9|a^2=i,c^2=h,d^2=ghi,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, 8324315);