Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2238 |
| automorphism | Aut(G) = #72313686 (order 4096) · Inn(G) = #481 · Out(G) = C2×#120 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #481 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| socle | C2 | C2×#481 |
| prime radical | #3054 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
278 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 278 conjugacy classes of subgroups — too many to draw. Only its 109 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,12)(4,9)(5,17)(6,21)(7,11)(8,15)(10,20)(13,27)(14,28)(16,26)(18,32)(19,25)(22,29)(23,30)(24,31);(5,18)(10,26)(12,28)(15,31)(16,20)(17,32)(19,30)(27,29);(1,3,6,22,7,14,4,13)(2,8,9,23,11,24,21,25)(5,19,20,31,18,30,16,15)(10,27,17,28,26,29,32,12);(1,4,7,6)(2,9,11,21)(3,13,14,22)(5,16,18,20)(8,23,24,25)(10,17,26,32)(12,27,28,29)(15,30,31,19)Polycyclic presentation:
2^8|c^2=eg,d^2=h,f^2=gh,g^2=h,b^a=be,c^a=cf,f^a=fg,g^a=gh,f^b=fh,e^c=eh,f^c=fgh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #137445, #137448, #137453, … |
| maximal normal subgroup of | #137445, #137448, #137453, … |
| maximal quotient of | #493919, #494269, #494485, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26644);