Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6355853 |
| 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 | #62117 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ 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:
(1,2,6,10)(3,11,15,25)(4,8,16,22)(5,18,12,9)(7,19,21,14)(13,24,27,32)(17,29,26,23)(20,30,31,28)(33,34,39,44)(35,40,46,54)(36,49,51,58)(37,43,38,42)(41,52,57,50)(45,59,60,64)(47,56,48,55)(53,62,63,61);(1,3,12,19,6,15,5,14)(2,7,18,25,10,21,9,11)(4,13,26,30,16,27,17,28)(8,20,29,32,22,31,23,24)(33,35)(34,40)(36,45)(37,47)(38,48)(39,46)(41,53)(42,55)(43,56)(44,54)(49,59)(50,61)(51,60)(52,62)(57,63)(58,64);(33,36,39,51)(34,41,44,57)(35,45,46,60)(37,50,38,52)(40,53,54,63)(42,49,43,58)(47,61,48,62)(55,59,56,64);(1,4)(2,8)(3,13)(5,17)(6,16)(7,20)(9,23)(10,22)(11,24)(12,26)(14,28)(15,27)(18,29)(19,30)(21,31)(25,32)(35,46)(40,54)(45,60)(47,48)(53,63)(55,56)(59,64)(61,62);(4,16)(8,22)(13,27)(17,26)(20,31)(23,29)(24,32)(28,30)(33,37,39,38)(34,42,44,43)(35,47,46,48)(36,50,51,52)(40,55,54,56)(41,49,57,58)(45,61,60,62)(53,59,63,64)Polycyclic presentation:
2^9|a^2=hi,b^2=fh,c^2=i,e^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fh,g^a=gi,d^b=di,g^c=gi,e^d=eh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6263116);