Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13175 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1943 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C2 | #10863 |
| 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 | |
503 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)(3,15,19,35)(4,22,25,57)(5,12)(6,30)(7,33)(8,14)(9,17,36,18)(10,24,40,60)(11,28)(13,32)(16,47)(20,46,53,38)(21,55,52,37)(23,58)(26,42,31,62)(27,41,61,43)(29,44)(34,59)(39,51)(45,48)(49,63)(50,64)(54,56);(1,3,6,20,32,52,29,18,8,19,28,53,7,21,5,17)(2,9,12,37,33,46,11,15,14,36,44,55,13,38,30,35)(4,23,26,49,61,16,24,50,25,59,31,48,27,51,60,54)(10,39,41,45,62,34,57,64,40,47,43,63,42,58,22,56);(1,4,5,24,7,27,28,26,8,25,29,60,32,61,6,31)(2,10,11,22,13,42,12,43,14,40,30,57,33,62,44,41)(3,16,18,49,21,23,20,54,19,51,17,48,52,59,53,50)(9,34,35,63,38,39,55,64,36,58,15,45,46,47,37,56)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=e,d^2=h,e^2=h,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gh,h^a=hi,c^b=ch,e^b=ei,f^b=fi,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72307189, #72307202, #72307253, … |
| maximal normal subgroup of | #72307189, #72307202, #72307253, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54302);