Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #7529 |
| automorphism | Aut(G) = #72286925 (order 4096) · Inn(G) = #1408 · Out(G) = C2×C22⋊C4 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1408 |
| commutator | C4×C8 | C2×C4 |
| Frattini, prime radical, cosocle | #493 | C22 |
| LCS | G ▷ C4×C8 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C42 ◁ #493 ◁ G | |
| LECS | G ▷ #493 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
76 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2)(3,11,13,28)(4,15,18,25)(5,10)(6,9,17,27)(7,12,21,31)(8,19,24,30)(14,29,20,23)(16,26)(22,32);(1,3,5,14)(2,7,10,21)(4,16,19,32)(6,20,17,13)(8,23,25,28)(9,26,27,22)(11,15,29,24)(12,30,31,18)Polycyclic presentation:
2^8|a^2=d,b^2=h,c^2=fg,e^2=gh,f^2=h,b^a=bc,c^a=ce,e^a=ef,f^a=fg,g^a=gh,c^b=cfgh,d^b=def,e^b=egh,f^b=fh,d^c=dfg,e^d=egh,f^d=fh25 conjugacy classes — view the full character table.
| maximal subgroup of | #94821, #94827, #94828, … |
| maximal normal subgroup of | #94821, #94827, #94828, … |
| maximal quotient of | #94568, #94585, #94643, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 518);