Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6364250 |
| 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 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1528 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,6,9)(3,11,14,21)(4,8,17,23)(5,18,19,10)(7,13,22,15)(12,25,27,31)(16,29,30,24)(20,26,32,28)(33,34,39,43)(35,40,46,55)(36,48,50,56)(37,42,38,51)(41,52,57,49)(44,58,60,63)(45,54,47,61)(53,62,64,59);(1,3,6,14)(2,7,9,22)(4,12,17,27)(5,13,19,15)(8,20,23,32)(10,21,18,11)(16,26,30,28)(24,31,29,25)(33,35)(34,40)(36,44)(37,45)(38,47)(39,46)(41,53)(42,54)(43,55)(48,58)(49,59)(50,60)(51,61)(52,62)(56,63)(57,64);(1,4,6,17)(2,8,9,23)(3,12,14,27)(5,16,19,30)(7,20,22,32)(10,24,18,29)(11,25,21,31)(13,26,15,28)(33,36)(34,41)(35,44)(37,49)(38,52)(39,50)(40,53)(42,56)(43,57)(45,59)(46,60)(47,62)(48,51)(54,63)(55,64)(58,61);(3,13)(5,19)(7,21)(10,18)(11,22)(12,26)(14,15)(16,30)(20,31)(24,29)(25,32)(27,28)(33,37,39,38)(34,42,43,51)(35,45,46,47)(36,49,50,52)(40,54,55,61)(41,56,57,48)(44,59,60,62)(53,63,64,58);(2,9)(3,14)(8,23)(10,18)(12,27)(13,15)(24,29)(26,28)(35,46)(40,55)(44,60)(45,47)(53,64)(54,61)(58,63)(59,62)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=h,d^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=gi,d^b=df,e^b=ehi,f^b=fh,g^c=gi,f^d=fh65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6271613);