Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24274 |
| 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 | |
330 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)(11,14)(12,13)(15,16)(17,18)(19,24)(20,25)(21,26)(22,27)(23,29)(28,32)(30,31);(1,3,4,8)(2,5,6,7)(17,19,20,30,21,29,22,28)(18,23,25,32,26,24,27,31);(10,13)(14,16)(17,19,21,29)(18,24,26,23)(20,28,22,30)(25,32,27,31);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(17,20,21,22)(18,25,26,27)(19,28,29,30)(23,31,24,32)Polycyclic presentation:
2^8|b^2=eg,c^2=h,d^2=fh,g^2=h,b^a=be,c^a=cf,d^a=deh,c^b=cg,d^b=dh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#28868, #6459890, #6459896, … |
| maximal normal subgroup of | C2×#28868, #6459890, #6459896, … |
| maximal quotient of | #362381, #362440, #366066, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21856);