Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37436 |
| automorphism | Aut(G) order 393216 |
| 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 | |
914 conjugacy classes of subgroups; only the 405 normal subgroups are shown.
This group has 914 conjugacy classes of subgroups — too many to draw. Only its 405 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:
(1,2)(3,7)(4,5)(6,8)(9,10,13,15)(11,16,12,14)(18,22)(23,24);(1,2)(3,6)(4,5)(7,8)(9,11,13,12)(10,14,15,16);(2,5)(6,7)(10,15)(14,16)(17,18)(19,23)(20,24)(21,22);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,16)(13,15)(17,19,21,20)(18,23,22,24);(1,3,4,8)(2,6,5,7)(9,12,13,11)(10,16,15,14)(17,20,21,19)(18,24,22,23)Polycyclic presentation:
2^8|a^2=f,b^2=f,d^2=fgh,e^2=g,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^c=dfh52 conjugacy classes — view the full character table.
| Fitting subgroup of | #37611⋊C3 |
| maximal subgroup of | #97048, #131216, #131222, … |
| maximal normal subgroup of | #97048, #131216, #131222, … |
| maximal quotient of | #556510, #556512, #556744, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30599);