Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13291 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1993 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C2 | #10881 |
| 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 | |
497 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,6,13,9,15,8,33)(3,16,20,46,19,38,30,47)(4,11,24,42,26,43,25,59)(5,27,28,14,7,31,29,12)(10,34,37,18,36,52,39,21)(17,48,22,54,51,45,56,64)(23,57,49,44,32,60,58,41)(35,55,63,53,62,61,40,50);(1,3,7,21,8,20,28,34,9,19,5,18,6,30,29,52)(2,10,14,38,13,39,31,46,15,36,12,16,33,37,27,47)(4,22,23,55,25,51,58,53,26,56,32,61,24,17,49,50)(11,40,41,54,42,62,57,48,43,63,44,64,59,35,60,45);(1,4,6,24,9,26,8,25)(2,11,13,42,15,43,33,59)(3,17,20,22,19,51,30,56)(5,23,28,49,7,32,29,58)(10,35,37,63,36,62,39,40)(12,41,27,57,14,44,31,60)(16,45,46,64,38,48,47,54)(18,50,52,55,21,53,34,61)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,d^2=h,f^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gh,h^a=hi,c^b=ch,e^b=ei,f^b=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72307179, #72307245, #72307275, … |
| maximal normal subgroup of | #72307179, #72307245, #72307275, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54392);