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 | |
3975 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 56 points:
(1,2,6,4)(3,5,8,7)(9,10,15,13)(11,16,19,12)(14,21,23,17)(18,20,22,24)(26,31)(29,41)(32,35)(33,42)(37,51)(40,52)(44,47)(50,56);(1,3)(2,7)(4,5)(6,8)(9,11,15,19)(10,16,13,12)(14,18,23,22)(17,24,21,20)(28,39)(33,42)(36,49)(40,52)(44,47)(46,53)(50,56)(54,55);(1,2,6,4)(3,7,8,5)(9,12,15,16)(10,11,13,19)(14,20,23,24)(17,18,21,22)(25,26)(27,35)(28,33)(29,34)(30,31)(32,38)(36,47)(37,48)(39,42)(40,46)(41,43)(44,49)(45,51)(50,55)(52,53)(54,56);(2,4)(3,8)(9,13,15,10)(11,16,19,12)(14,21,23,17)(18,24,22,20)(25,27,30,38)(26,32,31,35)(28,36,39,49)(29,37,41,51)(33,44,42,47)(34,45,43,48)(40,50,52,56)(46,54,53,55);(1,4,6,2)(3,7,8,5)(9,11)(10,16)(12,13)(14,22)(15,19)(17,20)(18,23)(21,24)(25,28)(26,33)(27,36)(29,40)(30,39)(31,42)(32,44)(34,46)(35,47)(37,50)(38,49)(41,52)(43,53)(45,54)(48,55)(51,56);(1,5,6,7)(2,3,4,8)(9,14)(10,17)(11,18)(12,20)(13,21)(15,23)(16,24)(19,22)(25,29,30,41)(26,34,31,43)(27,37,38,51)(28,40,39,52)(32,45,35,48)(33,46,42,53)(36,50,49,56)(44,54,47,55)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=gi,d^2=hi,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dgi,e^b=eghi,d^c=dghi,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10227585);