Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #121 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
455 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 455 conjugacy classes of subgroups — too many to draw. Only its 127 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(5,6,7,8)(10,14)(11,18)(12,15)(17,23)(20,21)(22,24);(1,2)(3,4)(9,10,15,23,13,17,12,14)(11,19,18,22,21,16,20,24);(2,4)(9,11)(10,16)(12,20)(13,21)(14,22)(15,18)(17,19)(23,24)Polycyclic presentation:
2^8|a^2=g,b^2=dh,d^2=h,e^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=cf,e^c=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #94283, #94284, #131276, … |
| maximal normal subgroup of | #94283, #94284, #131276, … |
| maximal quotient of | #116432, #116455, #116461, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4095);