Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #7535 |
| automorphism | Aut(G) = #72286283 (order 2048) · Inn(G) = #1409 · Out(G) = C22⋊C4 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1409 |
| 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 | |
69 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)(3,12,14,28)(4,16,18,8)(5,10,21,9)(6,11)(7,13,22,30)(15,29,20,25)(17,23)(19,26,32,27)(24,31);(1,3)(2,7,11,22)(4,17)(5,20)(6,15)(8,25,27,28)(9,24,10,23)(12,26,29,16)(13,18)(14,21)(19,31)(30,32)Polycyclic presentation:
2^8|a^2=d,b^2=gh,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=cfgh,d^b=defh,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 | C2×#7536, #153533, #153534, … |
| maximal normal subgroup of | C2×#7536, #153533, #153534, … |
| maximal quotient of | #94572, #94589, #94655, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 524);