Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#120 |
| automorphism | Aut(G) order 786432 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
795 conjugacy classes of subgroups; only the 503 normal subgroups are shown.
This group has 795 conjugacy classes of subgroups — too many to draw. Only its 503 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) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,6,4)(3,7,5,8)(11,15)(13,16)(17,18,20,22)(19,21,23,24);(1,2,6,4)(3,8,5,7)(9,10,12,14)(11,13,15,16)(17,18)(19,21)(20,22)(23,24);(1,3,6,5)(2,7,4,8)(9,11)(10,13)(12,15)(14,16)(17,19,20,23)(18,21,22,24);(1,4,6,2)(3,7,5,8);(1,5,6,3)(2,8,4,7)Polycyclic presentation:
2^8|a^2=fh,b^2=gh,c^2=fh,d^2=h,e^2=h,b^a=bf,c^a=cg,c^b=ch,e^b=eh,d^c=dh,e^d=eh70 conjugacy classes — view the full character table.
| Fitting subgroup of | #36584⋊C3 |
| maximal subgroup of | #98268, #224605, #224620, … |
| maximal normal subgroup of | #98268, #224605, #224620, … |
| maximal quotient of | #519000, #519060, #519443, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29572);