Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #94754 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C1 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7463 |
| commutator | C32 | C2×C8 |
| Frattini, cosocle | #1397 | C22 |
| prime radical | C22 | #1426 |
| LCS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ C4×C16 ◁ #1397 ◁ G | |
| LECS | G ▷ #1397 ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
46 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(2,8,10,28,12,30,14,38)(3,15,17,47)(4,20)(5,24)(6,25)(9,32,34,54)(11,33,39,56,41,63,43,64)(13,45,40,62,44,60,46,31)(16,49)(18,36,52,37)(19,35,53,48)(21,26)(22,51)(23,50)(27,55,42,29,57,59,58,61);(1,2,9,33,23,41,3,8,21,40,47,61,16,27,32,60,6,14,36,56,49,39,19,28,5,13,35,55,50,58,18,31,7,12,34,63,4,11,17,30,24,46,15,29,51,57,54,45,25,10,37,64,22,43,53,38,26,44,48,59,20,42,52,62)Polycyclic presentation:
2^9|a^2=d,b^2=ceg,c^2=egh,d^2=f,e^2=ghi,g^2=hi,h^2=i,b^a=bc,c^a=ce,e^a=eg,g^a=gh,h^a=hi,d^b=dh,f^b=fi,d^c=di56 conjugacy classes — view the full character table.
| maximal subgroup of | #72224652, #72225485, #72225500, … |
| maximal normal subgroup of | #72224652, #72225485, #72225500, … |
| maximal quotient of | #72230705, C2×#94754, C3×#94754, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1950);