Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | Aut(G) = C2×#72236479 (order 2048) · Inn(G) = D4 · Out(G) = C22×#525 |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C16 | D4 |
| commutator | C4 | C2×C4×C8 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C22 | C8×D4 |
| prime radical | C2×#41 | C2×C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C16 ◁ C22×C16 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
213 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 213 conjugacy classes of subgroups — too many to draw. Only its 111 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,9,15,10,14,22,29,11,16,23,30,8,12,21,25)(7,13,18,27,19,26,32,35,20,28,33,36,17,24,31,34);(2,4)(6,12,16,14)(13,24,28,26)(15,25,30,29)(27,34,36,35);(5,7,9,18,10,19,22,32,11,20,23,33,8,17,21,31)(6,13,15,27,14,26,29,35,16,28,30,36,12,24,25,34)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=e,d^2=h,e^2=g,g^2=h,b^a=bd,d^a=dh,f^a=fh112 conjugacy classes — view the full character table.
| maximal subgroup of | #151900, #151901, #152415, … |
| maximal normal subgroup of | #151900, #151901, #152415, … |
| maximal quotient of | #128249, #128266, #128285, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5602);