Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6350061 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | #63038 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2558 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,7,11)(3,12,15,22)(4,9,17,26)(5,10,6,19)(8,20,23,14)(13,27,30,32)(16,24,18,25)(21,28,31,29)(33,34)(35,40)(36,47)(37,43)(38,50)(39,42)(41,48)(44,59)(45,55)(46,54)(49,58)(51,56)(52,57)(53,62)(60,64)(61,63);(1,3,7,15)(2,8,11,23)(4,13,17,30)(5,14,6,20)(9,21,26,31)(10,22,19,12)(16,28,18,29)(24,27,25,32)(33,35)(34,40)(36,44)(37,45)(38,51)(39,46)(41,53)(42,54)(43,55)(47,63)(48,60)(49,61)(50,56)(52,62)(57,64)(58,59);(33,36,39,49)(34,41,42,57)(35,44,46,61)(37,48,38,52)(40,53,54,64)(43,58,50,47)(45,60,51,62)(55,59,56,63);(1,4)(2,9)(3,13)(5,18)(6,16)(7,17)(8,21)(10,25)(11,26)(12,27)(14,29)(15,30)(19,24)(20,28)(22,32)(23,31)(34,42)(40,54)(41,57)(43,50)(47,58)(53,64)(55,56)(59,63);(1,5,7,6)(2,10,11,19)(3,14,15,20)(4,16,17,18)(8,22,23,12)(9,24,26,25)(13,28,30,29)(21,27,31,32)(33,37,39,38)(34,43,42,50)(35,45,46,51)(36,48,49,52)(40,55,54,56)(41,58,57,47)(44,60,61,62)(53,59,64,63)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=i,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fh,g^a=gi,f^b=fh,g^b=gi,f^c=fi,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, 6257580);