Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #7527 |
| automorphism | Aut(G) = #72286283 (order 2048) · Inn(G) = #1408 · Out(G) = C22⋊C4 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1408 |
| commutator | C4×C8 | C2×C4 |
| Frattini, cosocle | #492 | C22 |
| prime radical | #127 | D4 |
| LCS | G ▷ C4×C8 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C42 ◁ #492 ◁ G | |
| LECS | G ▷ #492 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
71 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)(3,12,14,25)(4,16,18,8)(5,10,21,9)(6,11)(7,13,22,30)(15,29,20,28)(17,24)(19,26,32,27)(23,31);(1,3,6,15)(2,7,11,22)(4,17,19,31)(5,20,21,14)(8,25,27,28)(9,24,10,23)(12,16,29,26)(13,18,30,32)Polycyclic presentation:
2^8|a^2=d,b^2=h,c^2=fgh,e^2=gh,f^2=h,b^a=bc,c^a=ce,e^a=ef,f^a=fg,g^a=gh,c^b=cfg,d^b=defh,e^b=egh,f^b=fh,d^c=dfg,e^d=egh,f^d=fh19 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7528, #153519, #153520, … |
| maximal normal subgroup of | C2×#7528, #153519, #153520, … |
| maximal quotient of | #94566, #94586, #94643, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 516);