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 |
| prime radical | #3515 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1444 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,7,12)(3,13,18,24)(4,9,20,27)(5,11,6,10)(8,16,25,17)(14,29,32,23)(15,30,31,22)(19,28,21,26)(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);(1,3,7,18)(2,8,12,25)(4,14,20,32)(5,17,6,16)(9,22,27,30)(10,13,11,24)(15,21,31,19)(23,28,29,26)(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)(2,9)(3,14)(5,21)(6,19)(7,20)(8,22)(10,28)(11,26)(12,27)(13,29)(15,16)(17,31)(18,32)(23,24)(25,30)(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);(1,4)(2,9)(3,15)(5,19)(6,21)(7,20)(8,23)(10,26)(11,28)(12,27)(13,30)(14,16)(17,32)(18,31)(22,24)(25,29);(1,5,7,6)(2,10,12,11)(3,16,18,17)(4,19,20,21)(8,24,25,13)(9,26,27,28)(14,31,32,15)(22,29,30,23)(34,41)(35,44)(40,55)(42,49)(43,58)(45,60)(46,56)(59,62)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=i,e^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, 6275453);