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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
948 conjugacy classes of subgroups; only the 397 normal subgroups are shown.
This group has 948 conjugacy classes of subgroups — too many to draw. Only its 397 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:
(1,2,5,3)(4,7,8,6)(10,13)(11,12)(19,23)(22,24);(2,3)(6,7)(9,10,14,13)(11,16,12,15)(18,21)(22,24);(9,11,14,12)(10,15,13,16)(17,18)(19,22)(20,21)(23,24);(1,3,5,2)(4,7,8,6)(9,12,14,11)(10,16,13,15)(17,19)(18,22)(20,23)(21,24);(1,4)(2,6)(3,7)(5,8)(9,13,14,10)(11,16,12,15)(17,18)(19,22)(20,21)(23,24)Polycyclic presentation:
2^8|a^2=fg,b^2=g,c^2=g,d^2=f,e^2=g,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 | #53303⋊C3 |
| maximal subgroup of | #97300, #179229, #179230, … |
| maximal normal subgroup of | #97300, #179229, #179230, … |
| maximal quotient of | #606733, #606945, #607055, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 46291);