Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6367198 |
| 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 | #3483 | C22 |
| 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:
(1,2,7,12)(3,13,14,24)(4,9,20,27)(5,11,6,10)(8,16,22,17)(15,29,30,25)(18,32,31,23)(19,28,21,26)(33,34,39,44)(35,40,47,55)(36,48,50,58)(37,43,38,42)(41,52,56,49)(45,59,62,63)(46,57,51,54)(53,60,64,61);(1,3,7,14)(2,8,12,22)(4,18,20,31)(5,17,6,16)(9,25,27,29)(10,13,11,24)(15,21,30,19)(23,28,32,26)(33,35)(34,40)(36,45)(37,51)(38,46)(39,47)(41,53)(42,57)(43,54)(44,55)(48,59)(49,61)(50,62)(52,60)(56,64)(58,63);(3,14)(8,22)(13,24)(15,30)(16,17)(18,31)(23,32)(25,29)(33,36,39,50)(34,41,44,56)(35,45,47,62)(37,49,38,52)(40,53,55,64)(42,48,43,58)(46,60,51,61)(54,63,57,59);(1,4,7,20)(2,9,12,27)(3,15,14,30)(5,19,6,21)(8,23,22,32)(10,26,11,28)(13,29,24,25)(16,18,17,31);(1,5,7,6)(2,10,12,11)(3,16,14,17)(4,19,20,21)(8,24,22,13)(9,26,27,28)(15,18,30,31)(23,25,32,29)(33,37,39,38)(34,42,44,43)(35,46,47,51)(36,49,50,52)(40,54,55,57)(41,48,56,58)(45,60,62,61)(53,63,64,59)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=i,d^2=h,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,f^b=fh,g^c=gi,f^d=fh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6274674);