Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12778 |
| automorphism | Aut(G) = #72256403 (order 2048) · Inn(G) = #1790 · Out(G) = C24 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1790 |
| commutator | C2×Q8 | C22×C4 |
| Frattini, cosocle | C22×Q8 | C23 |
| prime radical | #3544 | C2 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
365 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 365 conjugacy classes of subgroups — too many to draw. Only its 69 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:
(1,2,7,13,8,14,6,12)(3,15,17,31,18,29,23,25)(4,19,21,10,5,11,20,28)(9,24,26,16,27,32,30,22);(1,3)(2,9)(4,16)(5,22)(6,23)(7,17)(8,18)(10,25)(11,29)(12,26)(13,30)(14,27)(15,19)(20,24)(21,32)(28,31);(1,4,8,5)(2,10,14,28)(3,16,18,22)(6,21,7,20)(9,25,27,31)(11,13,19,12)(15,26,29,30)(17,24,23,32)Polycyclic presentation:
2^8|a^2=f,c^2=h,d^2=h,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,d^b=dh,e^b=eg,f^b=fgh,d^c=dgh,e^c=eh,f^c=fh,e^d=eh,f^d=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #152565, #152596, #152597, … |
| maximal normal subgroup of | #152565, #152596, #152597, … |
| maximal quotient of | #129587, #129589, #129590, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5770);