Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24390 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
484 conjugacy classes of subgroups; only the 131 normal subgroups are shown.
This group has 484 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:
(5,6,8,9)(7,11,12,10)(13,14,19,21)(15,20,24,28)(16,18,25,17)(22,23,27,26);(1,2)(3,4)(6,9)(10,11)(13,15)(14,20)(16,22)(17,26)(18,23)(19,24)(21,28)(25,27);(2,4)(13,16,18,21,19,25,17,14)(15,22,23,28,24,27,26,20);(5,7,8,12)(6,10,9,11)(13,17,19,18)(14,16,21,25)(15,23,24,26)(20,27,28,22)Polycyclic presentation:
2^8|a^2=eh,c^2=f,d^2=eh,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,d^b=dh,d^c=dh46 conjugacy classes — view the full character table.
| maximal subgroup of | #139441, #139443, #347180, … |
| maximal normal subgroup of | #139441, #139443, #347180, … |
| maximal quotient of | #295725, #295728, #295731, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18674);