Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12465 |
| automorphism | Aut(G) = #72309547 (order 1024) · Inn(G) = #1677 · Out(G) = C2×C4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1677 |
| commutator, Frattini, cosocle | C4×C8 | C23 |
| prime radical | #2213 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
98 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(2,8)(3,13)(4,9)(5,20)(6,19)(11,31)(12,21)(14,15)(16,18)(17,25)(22,27)(23,24)(26,32)(28,29);(1,2,9,24,6,11,5,21,7,12,20,31,19,23,4,8)(3,10,25,27,15,28,14,26,17,30,13,32,16,29,18,22);(1,3)(2,10)(4,18)(5,14)(6,15)(7,17)(8,22)(9,25)(11,28)(12,30)(13,20)(16,19)(21,26)(23,29)(24,27)(31,32)Polycyclic presentation:
2^8|b^2=dg,d^2=fh,e^2=gh,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=df,e^a=eg,f^a=fh,g^a=gh,e^b=eh,d^c=dh,e^c=eg,g^c=gh31 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#12465, #490558, #490559, … |
| maximal normal subgroup of | C2×#12465, #490558, #490559, … |
| maximal quotient of | #134990, #138573, #138667, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5453);