Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D4 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
401 conjugacy classes of subgroups; only the 167 normal subgroups are shown.
This group has 401 conjugacy classes of subgroups — too many to draw. Only its 167 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) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,3,4)(5,6)(7,10)(8,12)(9,11)(13,14,19,24,18,21,17,23)(15,25,16,20,27,28,26,22);(5,7,9,8)(6,10,11,12)(13,15,18,27)(14,20,21,22)(16,19,26,17)(23,28,24,25);(5,8,9,7)(6,10,11,12)(14,21)(20,22)(23,24)(25,28);(13,16,17,15,18,26,19,27)(14,22,23,28,21,20,24,25)Polycyclic presentation:
2^8|a^2=g,b^2=f,c^2=fh,d^2=e,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,d^b=de,e^b=eh,g^b=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #221159, #221162, #221188, … |
| maximal normal subgroup of | #221159, #221162, #221188, … |
| maximal quotient of | #226781, #226796, #226810, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15800);