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 | C2×#537 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4111 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 56 points:
(1,2,4,6)(3,5,7,8)(9,10,15,19)(11,13,22,16)(12,23,21,17)(14,24,20,18)(25,26,31,32)(27,29,34,41)(28,33,40,46)(30,35,45,47)(36,42,49,38)(37,43,44,50)(39,48,54,56)(51,55,53,52);(3,7)(5,8)(9,11,15,22)(10,16,19,13)(12,14,21,20)(17,24,23,18)(26,32)(29,41)(33,46)(35,47)(38,42)(43,50)(48,56)(52,55);(1,2)(3,5)(4,6)(7,8)(9,12,15,21)(10,17,19,23)(11,20,22,14)(13,24,16,18)(25,27,31,34)(26,29,32,41)(28,36,40,49)(30,44,45,37)(33,42,46,38)(35,50,47,43)(39,53,54,51)(48,52,56,55);(9,13)(10,11)(12,24)(14,23)(15,16)(17,20)(18,21)(19,22)(25,28)(26,33)(27,36)(29,42)(30,39)(31,40)(32,46)(34,49)(35,48)(37,51)(38,41)(43,55)(44,53)(45,54)(47,56)(50,52);(1,3)(2,5)(4,7)(6,8)(9,11)(10,13)(12,20)(14,21)(15,22)(16,19)(17,24)(18,23)(25,29)(26,34)(27,32)(28,38)(30,43)(31,41)(33,36)(35,44)(37,47)(39,52)(40,42)(45,50)(46,49)(48,51)(53,56)(54,55);(1,2)(3,5)(4,6)(7,8)(9,14,15,20)(10,18,19,24)(11,21,22,12)(13,23,16,17)(25,30,31,45)(26,35,32,47)(27,37,34,44)(28,39,40,54)(29,43,41,50)(33,48,46,56)(36,51,49,53)(38,52,42,55)Polycyclic presentation:
2^9|a^2=ghi,b^2=i,c^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,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, 8833714);