Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24274 |
| 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×D8 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
417 conjugacy classes of subgroups; only the 131 normal subgroups are shown.
This group has 417 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)(13,14,18,22)(15,23,25,20)(16,19,26,28)(17,27,24,21);(1,2)(3,4)(6,9)(10,11)(13,15,17,16,18,25,24,26)(14,19,21,23,22,28,27,20);(2,4)(13,15,24,26,18,25,17,16)(14,19,27,20,22,28,21,23);(5,7,8,12)(6,10,9,11)(13,16,17,25,18,26,24,15)(14,20,21,19,22,23,27,28)Polycyclic presentation:
2^8|a^2=ef,b^2=f,c^2=fh,d^2=e,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,c^b=cg,d^b=dh46 conjugacy classes — view the full character table.
| maximal subgroup of | #504174, #504794, #504925, … |
| maximal normal subgroup of | #504174, #504794, #504925, … |
| maximal quotient of | #309179, #309185, #309191, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 19480);