Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13561 |
| automorphism | Aut(G) = #72286298 (order 2048) · Inn(G) = C2×AΣL(1,𝔽16) · Out(G) = C24 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×AΣL(1,𝔽16) |
| commutator | C2×D4 | C22×C4 |
| Frattini, cosocle | #124 | C23 |
| prime radical | C2×H(5,𝔽2) | C4 |
| LCS | G ▷ C2×D4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ C2×#124 ◁ G | |
| LECS | G ▷ #124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
248 conjugacy classes of subgroups; only the 45 normal subgroups are shown.
This group has 248 conjugacy classes of subgroups — too many to draw. Only its 45 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 — 3 generators on 32 points:
(1,2,4,9,6,11,18,27)(3,12,14,25,13,28,29,32)(5,10,21,26,20,8,17,24)(7,16,23,19,22,30,31,15);(1,3)(2,7)(4,16)(5,19)(6,13)(8,12)(9,25)(10,28)(11,22)(14,21)(15,20)(17,29)(18,30)(23,26)(24,31)(27,32);(3,13)(7,22)(12,28)(14,29)(15,19)(16,30)(23,31)(25,32)Polycyclic presentation:
2^8|a^2=e,e^2=h,f^2=h,b^a=bd,d^a=df,f^a=fg,g^a=gh,c^b=ch,e^b=efh,f^b=fh,g^b=gh,e^d=egh,f^d=fh,f^e=fh28 conjugacy classes — view the full character table.
| maximal subgroup of | #152019, #152020, #152179, … |
| maximal normal subgroup of | #152019, #152020, #152179, … |
| maximal quotient of | #149539, #149643, #149676, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6553);