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×#121 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3973 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)(3,7)(4,5)(6,8)(10,13)(11,17)(16,20)(19,23)(25,26,31,36)(27,32,34,29)(28,33,42,48)(30,35,44,50)(37,45,39,40)(38,46,49,43)(41,47,54,56)(51,55,52,53);(2,5)(6,7)(9,10,14,13)(11,18,17,15)(12,20,21,16)(19,22,23,24)(27,34)(29,32)(30,44)(35,50)(37,39)(40,45)(41,54)(47,56);(1,2)(3,6)(4,5)(7,8)(9,11,14,17)(10,15,13,18)(12,19,21,23)(16,22,20,24)(25,27,31,34)(26,32,36,29)(28,39,42,37)(30,38,44,49)(33,40,48,45)(35,46,50,43)(41,52,54,51)(47,53,56,55);(1,3,4,8)(2,6,5,7)(9,10,14,13)(11,15,17,18)(12,16,21,20)(19,22,23,24)(25,28)(26,33)(27,37)(29,40)(30,41)(31,42)(32,45)(34,39)(35,47)(36,48)(38,51)(43,53)(44,54)(46,55)(49,52)(50,56);(9,12,14,21)(10,16,13,20)(11,19,17,23)(15,22,18,24)(25,29,31,32)(26,34,36,27)(28,40,42,45)(30,43,44,46)(33,39,48,37)(35,49,50,38)(41,53,54,55)(47,52,56,51);(9,13,14,10)(11,18,17,15)(12,20,21,16)(19,24,23,22)(25,30)(26,35)(27,38)(28,41)(29,43)(31,44)(32,46)(33,47)(34,49)(36,50)(37,51)(39,52)(40,53)(42,54)(45,55)(48,56)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=ghi,d^2=g,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8664609);