Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
413 conjugacy classes of subgroups; only the 131 normal subgroups are shown.
This group has 413 conjugacy classes of subgroups — too many to draw. Only its 131 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:
(2,3)(5,6)(7,9)(8,10)(11,12)(13,14,16,19)(15,21,23,18)(17,20,24,27)(22,28,25,26);(1,2,4,3)(5,7,8,11)(6,9,10,12)(15,22)(17,24)(18,26)(20,27)(21,28)(23,25);(6,10)(9,12)(13,15,16,23)(14,18,19,21)(17,25,24,22)(20,28,27,26);(1,3,4,2)Polycyclic presentation:
2^8|a^2=h,b^2=efh,c^2=h,d^2=e,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #279957, #279962, #279974, … |
| maximal normal subgroup of | #279957, #279962, #279974, … |
| maximal quotient of | #281029, #281030, #281031, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18005);