Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6368046 |
| 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 | |
1477 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)(3,13)(4,9)(5,10)(6,21)(7,11)(8,15)(12,23)(14,30)(16,26)(17,25)(18,32)(19,29)(20,27)(22,28)(24,31)(33,34)(35,39)(36,46)(37,49)(38,41)(40,47)(42,51)(43,57)(44,54)(45,61)(48,56)(50,55)(52,59)(53,60)(58,64)(62,63);(1,3,7,4)(2,8,11,25)(5,18,20,31)(6,17,23,15)(9,21,13,12)(10,26,27,30)(14,22,16,19)(24,29,32,28)(33,35,38,44)(34,39,41,54)(36,43,48,58)(37,45,51,60)(40,52,55,63)(42,53,49,61)(46,57,56,64)(47,59,50,62);(1,4,7,3)(2,9,11,13)(5,16,20,14)(6,17,23,15)(8,21,25,12)(10,26,27,30)(18,19,31,22)(24,28,32,29)(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)(2,10,11,27)(3,14,4,16)(6,22,23,19)(8,24,25,32)(9,26,13,30)(12,29,21,28)(15,31,17,18);(2,11)(3,4)(10,27)(12,21)(14,16)(15,17)(18,31)(28,29)(34,41)(35,44)(40,55)(42,49)(43,58)(45,60)(46,56)(59,62)Polycyclic presentation:
2^9|b^2=hi,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=ehi,f^b=fh,e^c=eh,g^c=gi,f^d=fh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6275448);