Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13569 |
| automorphism | Aut(G) = #72236721 (order 2048) · Inn(G) = C2×AΣL(1,𝔽16) · Out(G) = C24 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×AΣL(1,𝔽16) |
| commutator | C2×Q8 | C22×C4 |
| Frattini, cosocle | #124 | C23 |
| prime radical | #3018 | C2 |
| LCS | G ▷ C2×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ G | |
| LECS | G ▷ #124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
241 conjugacy classes of subgroups; only the 45 normal subgroups are shown.
This group has 241 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)(3,13,16,23)(4,9)(5,20)(6,11,22,28)(7,12)(8,15,24,32)(10,21,30,19)(14,25,31,26)(17,27,18,29);(1,3)(2,8)(4,15)(5,14)(6,16)(7,18)(9,27)(10,29)(11,24)(12,26)(13,30)(17,22)(19,32)(20,23)(21,31)(25,28);(1,4,7,5)(2,9,12,20)(3,14,18,15)(6,19,22,21)(8,23,26,27)(10,28,30,11)(13,25,29,24)(16,31,17,32)Polycyclic presentation:
2^8|a^2=e,c^2=h,d^2=h,f^2=h,b^a=bd,d^a=df,f^a=fg,g^a=gh,c^b=ch,d^b=dh,e^b=ef,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 | #152025, #152026, #152194, … |
| maximal normal subgroup of | #152025, #152026, #152194, … |
| maximal quotient of | #149544, #149648, #149680, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6558);