Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6357668 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#127 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | #3511 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1728 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)(3,13,18,30)(4,9,5,10)(6,21,14,11)(8,22,25,17)(15,28,16,29)(19,27,20,26)(23,32,24,31)(33,34)(35,40)(36,48)(37,42)(38,52)(39,43)(41,50)(44,57)(45,54)(46,55)(47,61)(49,56)(51,62)(53,60)(58,64)(59,63);(1,3,14,22,7,18,6,17)(2,8,21,30,12,25,11,13)(4,15,20,32,5,16,19,31)(9,23,27,29,10,24,26,28)(33,35)(34,40)(36,44)(37,46)(38,47)(39,45)(41,53)(42,55)(43,54)(48,57)(49,59)(50,60)(51,58)(52,61)(56,63)(62,64);(1,4,7,5)(2,9,12,10)(3,15,18,16)(6,19,14,20)(8,23,25,24)(11,26,21,27)(13,28,30,29)(17,31,22,32)(33,36,38,50,39,51,37,49)(34,41,42,48,43,56,52,62)(35,44,47,60,45,58,46,59)(40,53,55,57,54,63,61,64);(1,5,7,4)(2,10,12,9)(3,16,18,15)(6,20,14,19)(8,24,25,23)(11,27,21,26)(13,29,30,28)(17,32,22,31)(35,45)(40,54)(44,58)(46,47)(53,63)(55,61)(57,64)(59,60);(4,5)(9,10)(15,16)(19,20)(23,24)(26,27)(28,29)(31,32)(33,37,39,38)(34,42,43,52)(35,46,45,47)(36,49,51,50)(40,55,54,61)(41,48,56,62)(44,59,58,60)(53,57,63,64)Polycyclic presentation:
2^9|a^2=h,b^2=fh,c^2=gh,d^2=h,e^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,d^b=di,e^c=eh,e^d=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6264911);