Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2002 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #120 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
381 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 381 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) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,3)(5,6,7,8)(9,10)(11,18)(12,16)(13,15)(14,19)(17,21)(20,24)(22,23);(1,2,4,3)(9,11,12,19)(10,14,16,18)(13,22,21,20)(15,24,17,23);(1,3,4,2)(10,15)(13,21)(14,23)(16,17)(18,24)(20,22)Polycyclic presentation:
2^8|a^2=g,b^2=d,c^2=dh,e^2=h,b^a=bd,c^a=ce,e^a=eh,f^a=fh,c^b=cf,e^b=eh,e^c=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #94323, #94327, #131374, … |
| maximal normal subgroup of | #94323, #94327, #131374, … |
| maximal quotient of | #120001, #120020, #120021, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4199);