Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29736 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
339 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,6)(4,5)(7,8)(9,10,12,14)(11,15,16,13)(17,18)(19,22)(20,24)(21,23)(25,29)(26,31)(27,30)(28,32);(2,5)(6,8)(17,19,20,26,21,27,28,25)(18,22,24,31,23,30,32,29);(1,3)(2,6)(4,7)(5,8)(9,11,12,16)(10,13,14,15)(19,25)(20,28)(22,29)(24,32)(26,27)(30,31);(3,7)(6,8)(18,23)(22,30)(24,32)(29,31)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=f,g^2=h,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^c=de,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#30299, #6433820, #6433826, … |
| maximal normal subgroup of | C2×#30299, #6433820, #6433826, … |
| maximal quotient of | #380777, #380779, #380807, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23287);