Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D4 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C23 | C4×D4 |
| prime radical | C24 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
327 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 327 conjugacy classes of subgroups — too many to draw. Only its 119 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)(5,6,8,7)(9,10,13,17,14,18,23,24)(11,15,19,21,20,12,16,22);(1,2,4,3)(5,7,8,6)(9,11)(10,15)(12,18)(13,19)(14,20)(16,23)(17,21)(22,24);(1,3,4,2)(6,7)(9,12)(10,16)(11,17)(13,22)(14,15)(18,19)(20,24)(21,23)Polycyclic presentation:
2^8|a^2=g,b^2=e,c^2=d,f^2=h,g^2=h,b^a=bd,c^a=ce,c^b=cf,f^b=fh,f^c=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | #92875, #136198, #136226, … |
| maximal normal subgroup of | #92875, #136198, #136226, … |
| maximal quotient of | #120601, #120602, #120613, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4246);