Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3186 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×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×D8 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
475 conjugacy classes of subgroups; only the 163 normal subgroups are shown.
This group has 475 conjugacy classes of subgroups — too many to draw. Only its 163 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:
(1,2,4,3)(5,6)(7,10)(8,11)(9,12)(13,14,16,18)(15,21,22,28)(17,25,23,20)(19,27,26,24);(1,3,4,2)(5,7,10,6)(8,11,9,12)(14,18)(15,22)(20,25)(24,27);(5,8,10,9)(6,11,7,12)(13,15,23,27,16,22,17,24)(14,19,25,28,18,26,20,21);(5,9,10,8)(6,11,7,12)(14,18)(19,26)(20,25)(21,28)Polycyclic presentation:
2^8|a^2=g,b^2=eg,c^2=ef,d^2=eh,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ce64 conjugacy classes — view the full character table.
| maximal subgroup of | #139122, #139123, C2×#22271, … |
| maximal normal subgroup of | #139122, #139123, C2×#22271, … |
| maximal quotient of | #219363, #219369, #219388, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15259);