Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2002 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C2×#120 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
541 conjugacy classes of subgroups; only the 183 normal subgroups are shown.
This group has 541 conjugacy classes of subgroups — too many to draw. Only its 183 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) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,7)(3,8,4,6)(9,10)(11,15)(12,17)(13,21)(14,18)(16,19)(20,24)(22,23);(1,3,5,4)(2,6,7,8)(9,11)(10,15)(12,19)(13,22)(14,20)(16,17)(18,24)(21,23);(1,4,5,3)(2,6,7,8)(9,11)(10,16)(12,19)(13,22)(14,20)(15,21)(17,24)(18,23);(9,12,14,13)(10,17,18,21)(11,19,20,22)(15,16,24,23)Polycyclic presentation:
2^8|a^2=e,b^2=e,c^2=e,d^2=h,f^2=h,b^a=be,c^a=cf,f^a=fh,g^a=gh,c^b=cg,f^b=fh,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #137109, #137110, #137118, … |
| maximal normal subgroup of | #137109, #137110, #137118, … |
| maximal quotient of | #241129, #241130, #241131, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16548);