Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | Aut(G) = #72285642 (order 1024) · Inn(G) = #117 · Out(G) = C25 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #117 |
| commutator | C2×C4 | C22×C8 |
| Frattini, cosocle | C2×C16 | C23 |
| socle | C2 | #2163 |
| prime radical | C23 | M32 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ #346 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
85 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,3,9,4,10,13,24,5,11,14,23,15,25,27,32,6,7,12,18,16,19,26,29,17,8,20,21,28,22,30,31);(2,7)(3,12)(8,11)(10,19)(13,26)(14,20)(22,25)(27,30);(2,8,7,11)(3,12)(9,23,18,21)(10,22,19,25)(13,26)(14,20)(24,32,29,31)(27,30)Polycyclic presentation:
2^8|a^2=e,c^2=d,e^2=f,f^2=g,g^2=h,b^a=bd,c^a=cg,d^a=dh,e^b=eh,e^c=eh64 conjugacy classes — view the full character table.
| maximal subgroup of | #94001, #94002, #94008, … |
| maximal normal subgroup of | #94001, #94002, #94008, … |
| maximal quotient of | #151881, #151893, #151981, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6630);