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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4124 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 56 points:
(10,15)(12,16)(17,23)(20,21)(25,26)(27,36)(28,29)(30,45)(31,32)(33,37)(34,43)(35,46)(38,39)(40,47)(41,48)(42,49)(44,55)(50,51)(52,56)(53,54);(1,2)(3,4)(5,6)(7,8)(9,10,14,15)(11,16,19,12)(13,17,22,23)(18,21,24,20)(26,32)(27,37)(29,34)(35,45)(38,41)(40,51)(44,49)(52,54);(2,6)(3,7)(9,11,14,19)(10,16,15,12)(13,18,22,24)(17,21,23,20)(25,27)(26,33)(28,41)(29,39)(30,40)(31,37)(32,36)(34,48)(35,47)(38,43)(42,54)(44,53)(45,50)(46,51)(49,56)(52,55);(3,7)(4,8)(9,12)(10,11)(13,21)(14,16)(15,19)(17,24)(18,23)(20,22)(25,28)(26,34)(27,38)(29,32)(30,42)(31,43)(33,39)(35,49)(36,48)(37,41)(40,52)(44,45)(46,55)(47,53)(50,56)(51,54);(1,3)(2,4)(5,7)(6,8)(9,11)(10,16)(12,15)(13,18)(14,19)(17,21)(20,23)(22,24)(25,29,31,34)(26,28,32,43)(27,39,37,48)(30,44,46,49)(33,41,36,38)(35,42,45,55)(40,53,51,56)(47,54,50,52);(1,4)(2,3)(5,8)(6,7)(9,13,14,22)(10,17,15,23)(11,18,19,24)(12,20,16,21)(25,30,31,46)(26,35,32,45)(27,40,37,51)(28,42,43,55)(29,44,34,49)(33,47,36,50)(38,52,41,54)(39,53,48,56)Polycyclic presentation:
2^9|b^2=gh,c^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,e^d=egi,f^d=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9532580);