Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2204 |
| automorphism | Aut(G) = #72234841 (order 2048) · Inn(G) = B(4,𝔽2) · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | B(4,𝔽2) |
| commutator | C2×Q8 | C22×C4 |
| Frattini, cosocle | C2×#41 | C23 |
| socle | C2 | #1889 |
| prime radical | #3018 | C2 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
316 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 316 conjugacy classes of subgroups — too many to draw. Only its 57 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:
(1,2,6,11)(3,12)(4,9,13,22)(5,10)(7,17)(8,20)(14,27)(15,24)(16,29,19,32)(18,26)(21,28)(23,30,25,31);(1,3,13,15,6,14,4,17)(2,7,22,12,11,24,9,27)(5,19,21,31,20,16,18,30)(8,25,26,32,10,23,28,29);(2,8)(3,14)(5,20)(7,23)(9,28)(10,11)(12,29)(15,17)(18,21)(22,26)(24,25)(27,32)Polycyclic presentation:
2^8|a^2=f,b^2=df,d^2=h,e^2=h,b^a=bd,c^a=ce,e^a=egh,g^a=gh,c^b=ch,d^b=dh,e^b=egh,f^b=fh,d^c=dg,e^c=eh,f^c=fg,e^d=eh,f^e=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #152180, #152181, #152188, … |
| maximal normal subgroup of | #152180, #152181, #152188, … |
| maximal quotient of | #136827, #136828, #136855, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6042);