Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6349901 |
| 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 | #62761 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2591 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2)(3,13)(4,9)(5,10)(6,11)(7,12)(8,15)(14,27)(16,22)(17,24)(18,26)(19,25)(20,23)(21,28)(29,32)(30,31)(33,34,39,42)(35,45,49,56)(36,50,53,58)(37,44,38,43)(40,47,55,48)(41,51,57,52)(46,60,63,54)(59,62,64,61);(1,3)(2,8)(4,14)(5,15)(6,20)(7,16)(9,21)(10,22)(11,13)(12,23)(17,28)(18,29)(19,30)(24,31)(25,32)(26,27)(33,35)(34,40)(36,46)(37,47)(38,48)(39,49)(41,54)(42,55)(43,56)(44,45)(50,64)(51,61)(52,62)(53,63)(57,60)(58,59);(1,4,7,19)(2,9,12,25)(3,14,16,30)(5,17,6,18)(8,21,23,32)(10,24,11,26)(13,27,22,31)(15,28,20,29)(33,36,39,53)(34,41,42,57)(35,46,49,63)(37,52,38,51)(40,54,55,60)(43,50,44,58)(45,59,56,64)(47,62,48,61);(1,5,7,6)(2,10,12,11)(3,15,16,20)(4,17,19,18)(8,22,23,13)(9,24,25,26)(14,28,30,29)(21,31,32,27)(34,42)(40,55)(41,57)(43,44)(45,56)(50,58)(54,60)(59,64);(1,5,7,6)(2,11,12,10)(3,15,16,20)(4,18,19,17)(8,13,23,22)(9,24,25,26)(14,29,30,28)(21,31,32,27)(33,37,39,38)(34,43,42,44)(35,47,49,48)(36,51,53,52)(40,56,55,45)(41,58,57,50)(46,61,63,62)(54,59,60,64)Polycyclic presentation:
2^9|a^2=i,c^2=h,d^2=hi,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fh,g^a=gi,f^b=fh,g^b=gi,e^c=eh,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6257142);