Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23860 |
| 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 ◁ C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
348 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)(3,7)(4,6)(5,8)(9,10,12,14)(11,15,16,13)(17,18,22,26)(19,23,29,32)(20,24,21,25)(27,30,28,31);(1,3,4,8)(2,5,6,7)(17,19)(18,23)(20,28)(21,27)(22,29)(24,31)(25,30)(26,32);(9,11,12,16)(10,13,14,15)(19,27)(20,21)(23,30)(24,25)(28,29)(31,32);(2,6)(5,7)(17,20,22,21)(18,24,26,25)(19,28,29,27)(23,31,32,30)Polycyclic presentation:
2^8|a^2=fh,b^2=e,c^2=f,d^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,d^c=dh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#26013, #6452312, #6452318, … |
| maximal normal subgroup of | C2×#26013, #6452312, #6452318, … |
| maximal quotient of | #360628, #364264, #365805, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 19001);