Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12933 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1853 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C2 | #10908 |
| prime radical | C24 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
513 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2)(3,15,19,45)(4,21,24,44)(5,26)(6,30)(7,34)(8,13)(9,17,39,31)(10,23,42,27)(11,28)(12,29)(14,33)(16,50)(18,40,55,47)(20,56,35,38)(22,43,32,48)(25,46,36,41)(37,57)(49,60)(51,61)(52,64)(53,62)(54,63)(58,59);(1,3)(2,9,13,39)(4,16)(5,18)(6,31)(7,35)(8,19)(10,37,42,62)(11,38,30,56)(12,45,26,15)(14,47,34,40)(17,28)(20,33)(21,58,44,61)(22,52)(23,59)(24,54)(25,57)(27,51)(29,55)(32,60)(36,53)(41,49,46,64)(43,50,48,63);(1,4,7,36,8,24,33,25)(2,10,14,48,13,42,34,43)(3,16,35,53,19,54,20,57)(5,27,28,22,29,23,6,32)(9,37,47,63,39,62,40,50)(11,44,26,41,30,21,12,46)(15,49,56,58,45,64,38,61)(17,52,55,59,31,60,18,51)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=fh,d^2=h,e^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,g^a=gi,h^a=hi,d^b=dh,e^b=eh,f^b=fi,h^b=hi,d^c=dh,e^c=ehi,f^c=fi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72307178, #72307236, #72307275, … |
| maximal normal subgroup of | #72307178, #72307236, #72307275, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54481);