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 | #63032 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2552 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)(3,12,15,21)(4,9,6,19)(5,10,18,26)(8,20,23,13)(14,27,30,31)(16,24,17,25)(22,29,32,28)(33,34,38,41)(35,39,45,54)(36,46,47,58)(37,49,52,42)(40,51,56,48)(43,59,60,63)(44,57,50,55)(53,62,64,61);(1,3)(2,8)(4,13)(5,14)(6,20)(7,15)(9,21)(10,22)(11,23)(12,19)(16,28)(17,29)(18,30)(24,31)(25,27)(26,32)(33,35,38,45)(34,39,41,54)(36,43,47,60)(37,50,52,44)(40,53,56,64)(42,57,49,55)(46,63,58,59)(48,62,51,61);(1,4,7,6)(2,9,11,19)(3,13,15,20)(5,16,18,17)(8,21,23,12)(10,24,26,25)(14,28,30,29)(22,31,32,27)(33,36)(34,40)(35,43)(37,51)(38,47)(39,53)(41,56)(42,58)(44,62)(45,60)(46,49)(48,52)(50,61)(54,64)(55,63)(57,59);(1,5)(2,10)(3,14)(4,16)(6,17)(7,18)(8,22)(9,24)(11,26)(12,27)(13,28)(15,30)(19,25)(20,29)(21,31)(23,32)(34,41)(39,54)(40,56)(42,49)(46,58)(53,64)(55,57)(59,63);(1,4,7,6)(2,9,11,19)(3,13,15,20)(5,17,18,16)(8,21,23,12)(10,25,26,24)(14,29,30,28)(22,27,32,31)Polycyclic presentation:
2^9|a^2=hi,b^2=i,c^2=h,e^2=h,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, 6257324);