Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| 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 | C23 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
383 conjugacy classes of subgroups; only the 167 normal subgroups are shown.
This group has 383 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)(7,9)(8,10)(11,12)(13,14,18,17,19,22,16,21)(15,23,24,20,26,27,25,28);(5,7,8,11)(6,9,10,12)(13,15,19,26)(14,20,22,28)(16,24,18,25)(17,23,21,27);(6,10)(9,12)(13,16,19,18)(14,21,22,17)(15,24,26,25)(20,27,28,23);(13,17,18,22,19,21,16,14)(15,20,24,27,26,28,25,23)Polycyclic presentation:
2^8|a^2=g,b^2=fh,c^2=h,d^2=eh,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 | #138989, #138991, #220546, … |
| maximal normal subgroup of | #138989, #138991, #220546, … |
| maximal quotient of | #225237, #225251, #225265, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15608);