Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×D8 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
367 conjugacy classes of subgroups; only the 203 normal subgroups are shown.
This group has 367 conjugacy classes of subgroups — too many to draw. Only its 203 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,3,4)(6,10)(11,12)(13,14)(15,18)(16,24)(17,21)(19,22)(20,23)(25,26)(27,28);(5,6,9,10)(7,11,8,12)(13,15,17,23)(14,18,21,20)(16,22,25,28)(19,26,27,24);(5,7,9,8)(6,11,10,12)(13,16,23,28,17,25,15,22)(14,19,18,26,21,27,20,24);(5,8,9,7)(6,11,10,12)Polycyclic presentation:
2^8|a^2=g,b^2=eh,c^2=efh,d^2=e,f^2=h,b^a=be,c^a=cf,f^a=fh,d^b=de70 conjugacy classes — view the full character table.
| maximal subgroup of | #135622, #135623, #135624, … |
| maximal normal subgroup of | #135622, #135623, #135624, … |
| maximal quotient of | #226427, #226435, #226437, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15729);