Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6356901 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | #62145 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1464 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)(3,12)(4,9)(5,10)(6,21)(7,11)(8,16)(13,28)(14,29)(15,22)(17,32)(18,25)(19,27)(20,26)(23,30)(24,31)(33,34)(35,45)(36,50)(37,42)(38,43)(39,44)(40,48)(41,52)(46,58)(47,57)(49,56)(51,55)(53,59)(54,60)(61,63)(62,64);(1,3,6,16,7,13,5,15)(2,8,10,12,11,22,21,28)(4,17,18,31,19,14,20,30)(9,24,26,32,27,23,25,29)(33,35,38,47,39,49,37,48)(34,40,42,56,44,57,43,45)(36,51,54,63,53,46,52,62)(41,58,59,61,60,55,50,64);(3,13)(8,22)(12,28)(14,17)(15,16)(23,24)(29,32)(30,31)(33,36,39,53)(34,41,44,60)(35,46,49,51)(37,52,38,54)(40,55,57,58)(42,59,43,50)(45,61,56,64)(47,62,48,63);(1,4,7,19)(2,9,11,27)(3,14,13,17)(5,20,6,18)(8,23,22,24)(10,26,21,25)(12,29,28,32)(15,31,16,30)(33,37,39,38)(34,42,44,43)(35,47,49,48)(36,52,53,54)(40,45,57,56)(41,59,60,50)(46,62,51,63)(55,61,58,64);(1,5,7,6)(2,10,11,21)(3,15,13,16)(4,18,19,20)(8,12,22,28)(9,25,27,26)(14,30,17,31)(23,32,24,29)(33,37,39,38)(34,42,44,43)(35,48,49,47)(36,52,53,54)(40,56,57,45)(41,59,60,50)(46,63,51,62)(55,64,58,61)Polycyclic presentation:
2^9|b^2=fi,c^2=i,d^2=h,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,c^b=ch,d^b=dh,g^c=gi,e^d=ehi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6264198);