Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6367534 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | #121 | C2×D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1432 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2)(3,11)(4,7)(5,9)(6,17)(8,14)(10,23)(12,21)(13,29)(15,27)(16,20)(18,30)(19,28)(22,26)(24,32)(25,31)(33,34,38,41)(35,39,44,54)(36,46,48,56)(37,49,51,42)(40,50,55,47)(43,57,58,64)(45,61,60,53)(52,62,63,59);(1,3,7,15)(2,8,4,12)(5,18,20,32)(6,21,23,14)(9,25,16,29)(10,27,17,11)(13,22,31,19)(24,28,30,26)(33,35)(34,39)(36,43)(37,45)(38,44)(40,52)(41,54)(42,53)(46,57)(47,59)(48,58)(49,61)(50,62)(51,60)(55,63)(56,64);(1,4,7,2)(3,12,15,8)(5,16,20,9)(6,17,23,10)(11,14,27,21)(13,30,31,24)(18,29,32,25)(19,28,22,26)(33,36,38,48)(34,40,41,55)(35,43,44,58)(37,50,51,47)(39,52,54,63)(42,56,49,46)(45,62,60,59)(53,64,61,57);(1,5,7,20)(2,9,4,16)(3,13,15,31)(6,22,23,19)(8,24,12,30)(10,28,17,26)(11,29,27,25)(14,32,21,18);(2,4)(8,12)(9,16)(10,17)(11,27)(24,30)(25,29)(26,28)(34,41)(35,44)(40,55)(42,49)(43,58)(45,60)(46,56)(59,62)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=hi,d^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fh,g^a=gi,d^b=df,e^b=ei,f^b=fh,e^c=eh,g^c=gi,f^d=fh59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6274840);