Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #199558 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#121 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
963 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2)(3,12)(4,16,17,26)(5,20)(6,10)(7,11)(8,15)(9,19,25,18)(13,28,30,22)(14,24)(21,23)(27,29,32,31)(33,34,38,42,39,43,37,52)(35,40,47,54,46,55,45,57)(36,48,51,58,50,41,49,56)(44,53,61,63,60,62,59,64);(1,3,5,15,7,14,6,21)(2,8,10,12,11,23,20,24)(4,13,19,31,17,30,18,29)(9,22,26,27,25,28,16,32)(33,35,38,45,39,46,37,47)(34,40,52,54,43,55,42,57)(36,44,49,61,50,60,51,59)(41,53,56,64,48,62,58,63);(1,4)(2,9)(3,13)(5,19)(6,18)(7,17)(8,22)(10,26)(11,25)(12,27)(14,30)(15,31)(16,20)(21,29)(23,28)(24,32)(33,36)(34,41)(35,44)(37,51)(38,49)(39,50)(40,53)(42,58)(43,48)(45,61)(46,60)(47,59)(52,56)(54,64)(55,62)(57,63);(3,14)(4,17)(8,23)(9,25)(12,24)(15,21)(16,26)(18,19)(33,37,39,38)(34,42,43,52)(35,45,46,47)(36,49,50,51)(40,54,55,57)(41,56,48,58)(44,59,60,61)(53,63,62,64)Polycyclic presentation:
2^9|a^2=g,b^2=eg,d^2=i,e^2=hi,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,d^b=dh,e^b=ei,f^b=fh,g^b=gi,d^c=dh,e^c=eh,f^c=fhi,g^c=gh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 106906);