Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| automorphism | Aut(G) order 524288 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1085 conjugacy classes of subgroups; only the 423 normal subgroups are shown.
This group has 1085 conjugacy classes of subgroups — too many to draw. Only its 423 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 — 5 generators on 24 points:
(4,8)(6,7)(9,10)(11,15)(12,13)(14,16)(17,18)(19,23)(20,21)(22,24);(1,2,5,3)(4,7,8,6)(9,11,13,16)(10,14,12,15)(18,21)(19,24);(2,3)(6,7)(11,16)(14,15)(17,18,20,21)(19,22,24,23);(1,3,5,2)(4,7,8,6)(9,11)(10,15)(12,14)(13,16)(17,18)(19,22)(20,21)(23,24);(1,4,5,8)(2,6,3,7)(9,12,13,10)(11,14,16,15)(17,19,20,24)(18,22,21,23)Polycyclic presentation:
2^8|b^2=gh,c^2=g,d^2=fh,e^2=h,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,d^c=dh58 conjugacy classes — view the full character table.
| maximal subgroup of | #130523, #132537, #253908, … |
| maximal normal subgroup of | #130523, #132537, #253908, … |
| maximal quotient of | #535422, #536134, #536540, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 38425);