Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13304 |
| automorphism | Aut(G) = C27×#442 (order 8192) · Inn(G) = C2×D8 · Out(G) = C26×C4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C2×D8 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
173 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 173 conjugacy classes of subgroups — too many to draw. Only its 65 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,3,4)(10,11)(12,13)(14,17)(15,24)(18,22)(19,20)(21,23);(5,6)(7,8)(9,10,13,18,15,20,17,23,16,21,14,19,24,22,12,11);(2,4)(6,8)(9,11,16,23)(10,17,21,12)(13,22,14,20)(15,19,24,18)Polycyclic presentation:
2^8|a^2=de,b^2=dg,c^2=h,d^2=gh,e^2=g,f^2=g,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=egh,f^a=fgh,g^a=gh,c^b=cf,d^c=dg,e^c=egh,f^c=fgh,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13475, #499324, #499325, … |
| maximal normal subgroup of | C2×#13475, #499324, #499325, … |
| maximal quotient of | #145901, #145911, #145921, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6463);