Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | Aut(G) order 12288 |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | C22×D4 |
| commutator | C4 | C23×C8 |
| Frattini, cosocle | C2×C8 | C24 |
| socle | C2 | C22×C4⋊C8 |
| prime radical | C2×#41 | C2×C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
342 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,12,7,11,17,27,8,13,16,9,5,22,3,14)(4,10,20,26,21,25,29,32,18,24,30,23,19,31,15,28);(1,3,5,16,8,17,7,6)(2,9,11,14,13,12,22,27)(4,15,19,30,18,29,21,20)(10,23,25,28,24,26,31,32);(1,4)(2,10)(3,15)(5,19)(6,20)(7,21)(8,18)(9,23)(11,25)(12,26)(13,24)(14,28)(16,30)(17,29)(22,31)(27,32);(4,18)(10,24)(15,29)(19,21)(20,30)(23,26)(25,31)(28,32)Polycyclic presentation:
2^8|a^2=f,b^2=e,e^2=h,f^2=g,g^2=h,b^a=be,e^a=eh,d^c=dh88 conjugacy classes — view the full character table.
| Fitting subgroup of | #33401⋊C3 |
| maximal subgroup of | #126722, #126723, #126850, … |
| maximal normal subgroup of | #126722, #126723, #126850, … |
| maximal quotient of | #483325, #483339, #483350, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26389);