Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #199546 |
| 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 | #3444 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
961 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)(4,16,19,26)(5,20)(6,10)(7,11)(8,15)(9,18,25,17)(13,28,29,22)(14,24)(21,23)(27,30,32,31)(33,34,38,42,39,43,37,51)(35,40,47,54,46,55,45,58)(36,48,49,56,50,41,52,57)(44,53,59,64,60,62,61,63);(1,3,5,15,7,14,6,21)(2,8,10,12,11,23,20,24)(4,13,18,31,19,29,17,30)(9,22,26,27,25,28,16,32)(33,35,38,45,39,46,37,47)(34,40,51,54,43,55,42,58)(36,44,52,59,50,60,49,61)(41,53,57,63,48,62,56,64);(1,4)(2,9)(3,13)(5,18)(6,17)(7,19)(8,22)(10,26)(11,25)(12,27)(14,29)(15,31)(16,20)(21,30)(23,28)(24,32)(33,36)(34,41)(35,44)(37,49)(38,52)(39,50)(40,53)(42,56)(43,48)(45,59)(46,60)(47,61)(51,57)(54,63)(55,62)(58,64);(3,14)(8,23)(12,24)(13,29)(15,21)(22,28)(27,32)(30,31)(33,37,39,38)(34,42,43,51)(35,45,46,47)(36,49,50,52)(40,54,55,58)(41,56,48,57)(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,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, 106886);