Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2202 |
| automorphism | Aut(G) = #72312045 (order 6144) · Inn(G) = B(4,𝔽2) · Out(G) = C22×S4 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×Q8 | C24 |
| socle | C2 | C2×B(4,𝔽2) |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ G | |
| LECS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
556 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 556 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)(3,13)(4,10)(5,18)(6,9)(7,12)(8,14)(11,22)(15,29)(16,24)(17,27)(19,26)(20,28)(21,30)(23,31)(25,32);(1,3)(2,8)(4,14)(5,19)(6,21)(7,16)(9,23)(10,24)(11,25)(12,26)(13,18)(15,22)(17,32)(20,31)(27,30)(28,29);(2,9)(6,22)(8,23)(10,27)(11,12)(13,29)(15,21)(17,20)(18,28)(24,30)(25,26)(31,32);(1,4,7,5)(2,10,12,18)(3,14,16,19)(6,17,22,20)(8,24,26,13)(9,27,11,28)(15,31,21,32)(23,30,25,29)Polycyclic presentation:
2^8|d^2=h,e^2=h,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,g^a=gh,e^b=eh,f^b=fgh,e^c=eg,f^c=fh,f^e=fh34 conjugacy classes — view the full character table.
| Fitting subgroup of | #33564⋊C3 |
| maximal subgroup of | #152553, #153131, #153173, … |
| maximal normal subgroup of | #152553, #153131, #153173, … |
| maximal quotient of | #484171, #484257, #484295, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26552);