Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #93875 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7098 |
| commutator | C82 | C2×C4 |
| Frattini, cosocle | C8×C16 | C22 |
| prime radical | C22 | #1357 |
| LCS | G ▷ C82 ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ C8×C16 ◁ G | |
| LECS | G ▷ C8×C16 ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
83 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,4,10,20,36,54,61,25,12,43,39,26,11,41,38,6,13,22,40,5,9,21,34,7,27,23,56,55,35,62,58)(3,14,16,45,47,32,52,59,28,31,57,37,24,33,19,51,15,42,46,63,17,44,48,60,18,50,49,64,53,8,29,30);(1,3)(2,8,13,33)(4,19)(5,24)(6,15)(7,28)(9,31,36,50)(10,37,40,64)(11,42,35,14)(12,44,27,32)(16,41)(17,26)(18,25)(20,53)(21,49)(22,29)(23,48)(30,58,51,38)(34,60,61,59)(39,45,56,63)(43,52)(46,62)(47,55)(54,57)Polycyclic presentation:
2^9|a^2=d,b^2=h,c^2=fhi,d^2=efh,e^2=gi,f^2=h,g^2=i,b^a=bc,c^a=ce,e^a=eg,f^a=fg,g^a=gi,h^a=hi,c^b=cf,d^b=defgi,e^b=eg,f^b=fh,g^b=gi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1071);