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 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1481 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)(35,40)(36,48)(37,42)(38,51)(39,43)(41,49)(44,58)(45,54)(46,55)(47,61)(50,56)(52,57)(53,59)(60,63)(62,64);(1,3)(2,8)(4,18)(5,17)(6,16)(7,14)(9,25)(10,13)(11,24)(12,22)(15,19)(20,31)(21,30)(23,26)(27,29)(28,32)(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);(3,14)(8,22)(13,24)(15,30)(16,17)(18,31)(23,32)(25,29)(33,36,39,50)(34,41,43,57)(35,44,46,60)(37,49,38,52)(40,53,55,64)(42,56,51,48)(45,59,47,62)(54,63,61,58);(1,4)(2,9)(3,15)(5,19)(6,21)(7,20)(8,23)(10,26)(11,28)(12,27)(13,29)(14,30)(16,18)(17,31)(22,32)(24,25)(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);(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)(34,43)(35,46)(41,57)(42,51)(44,60)(45,47)(48,56)(59,62)Polycyclic presentation:
2^9|a^2=h,c^2=i,d^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,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, 6274561);