Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33568 |
| automorphism | Aut(G) = C2×#72236702 (order 2048) · Inn(G) = C2×B(4,𝔽2) · Out(G) = C24 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×Q8 | C24 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ G | |
| LECS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
554 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 554 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)(3,14)(4,12)(5,11)(6,21)(7,23)(8,13)(9,15)(10,19)(16,24)(17,31)(18,29)(20,28)(22,27)(25,30)(26,32);(1,3)(2,9)(4,15)(5,17)(6,22)(7,16)(8,18)(10,24)(11,26)(12,29)(13,27)(14,21)(19,32)(20,30)(23,25)(28,31);(1,4,8,6)(2,10,13,28)(3,15,18,22)(5,19,7,20)(9,24,27,31)(11,21,23,12)(14,25,29,26)(16,30,17,32);(1,5,8,7)(2,11,13,23)(3,16,18,17)(4,19,6,20)(9,25,27,26)(10,21,28,12)(14,24,29,31)(15,30,22,32)Polycyclic presentation:
2^8|c^2=h,d^2=h,e^2=h,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,g^a=gh,d^b=dh,e^b=eh,f^b=fgh,e^c=eg,f^c=fh,f^e=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #499881, #499884, #499885, … |
| maximal normal subgroup of | #499881, #499884, #499885, … |
| maximal quotient of | #484180, #484184, #484186, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26560);