Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3585 |
| automorphism | Aut(G) order 196608 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C23×D4 |
| commutator | C4 | C24×C4 |
| Frattini, cosocle | C2×C4 | C25 |
| socle | C2 | C23×C4⋊C4 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1282 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,10,7,11,5,20)(3,12,16,22,14,26,15,8)(4,9,19,24,17,23,18,30)(13,25,29,31,27,32,28,21);(1,3,5,15,7,14,6,16)(2,8,10,12,11,22,20,26)(4,13,18,28,17,27,19,29)(9,21,24,25,23,31,30,32);(1,4)(2,9)(3,13)(5,18)(6,19)(7,17)(8,21)(10,24)(11,23)(12,25)(14,27)(15,28)(16,29)(20,30)(22,31)(26,32);(4,17)(9,23)(13,27)(18,19)(21,31)(24,30)(25,32)(28,29);(3,14)(8,22)(12,26)(13,27)(15,16)(21,31)(25,32)(28,29)Polycyclic presentation:
2^8|a^2=g,b^2=f,f^2=h,g^2=h,b^a=bf,f^a=fh,e^b=eh,d^c=dh82 conjugacy classes — view the full character table.
| Fitting subgroup of | #60113⋊C3 |
| maximal subgroup of | #368076, #368085, #368091, … |
| maximal normal subgroup of | #368076, #368085, #368091, … |
| maximal quotient of | #6344310, #6344323, #6344339, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53101);