Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53158 |
| automorphism | Aut(G) order 196608 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | #2679 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
869 conjugacy classes of subgroups; only the 395 normal subgroups are shown.
This group has 869 conjugacy classes of subgroups — too many to draw. Only its 395 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 — 5 generators on 24 points:
(1,2)(3,5)(4,7)(6,8)(9,10)(11,16)(12,15)(13,14)(17,18,20,22)(19,23,24,21);(2,3)(6,7)(10,13)(12,16)(17,18,20,22)(19,21,24,23);(1,2,5,3)(4,6,8,7)(9,11)(10,12)(13,16)(14,15)(17,18)(19,21)(20,22)(23,24);(1,3,5,2)(4,7,8,6)(9,12,14,16)(10,15,13,11)(17,19,20,24)(18,21,22,23);(1,4,5,8)(2,6,3,7)(9,13,14,10)(11,16,15,12)Polycyclic presentation:
2^8|a^2=fg,b^2=fg,c^2=fgh,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=dh,e^a=ef,c^b=ch,d^b=df,e^b=egh46 conjugacy classes — view the full character table.
| Fitting subgroup of | #53373⋊C3 |
| maximal subgroup of | #97308, #179405, #179406, … |
| maximal normal subgroup of | #97308, #179405, #179406, … |
| maximal quotient of | #606822, #607060, #607074, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 46361);