Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #300501 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2433 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | #121 | C2×D4 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
800 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2)(3,12)(4,17)(5,20)(6,19)(7,11)(8,15)(9,21)(10,22)(13,14)(16,27)(18,32)(23,28)(24,29)(25,30)(26,31)(33,34,39,43)(35,40,46,57)(36,47,49,60)(37,51,38,42)(41,53,59,50)(44,58,61,64)(45,56,52,55)(48,63,62,54);(1,3,6,15,7,16,22,30)(2,8,10,12,11,25,19,27)(4,18,20,26,21,13,28,29)(5,14,17,31,23,32,9,24)(33,35)(34,40)(36,48)(37,45)(38,52)(39,46)(41,58)(42,55)(43,57)(44,50)(47,63)(49,62)(51,56)(53,61)(54,60)(59,64);(1,4)(2,9)(3,13,16,18)(5,10)(6,20)(7,21)(8,24,25,31)(11,17)(12,14,27,32)(15,29,30,26)(19,23)(22,28)(33,36,37,50,39,49,38,53)(34,41,42,47,43,59,51,60)(35,44,45,62,46,61,52,48)(40,54,55,58,57,63,56,64);(1,5,22,9,7,23,6,17)(2,4,19,28,11,21,10,20)(3,14,30,24,16,32,15,31)(8,18,27,29,25,13,12,26)(36,49)(41,59)(44,61)(47,60)(48,62)(50,53)(54,63)(58,64)Polycyclic presentation:
2^9|a^2=i,b^2=e,c^2=fh,d^2=eh,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gh,c^b=cg,f^b=fh,g^b=gi,d^c=dhi,g^c=ghi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 207711);