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 | C22×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
337 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,6)(3,7,4,8)(9,10)(11,14)(12,16)(13,15)(17,18)(19,26)(20,30)(21,24)(22,25)(23,29)(27,31)(28,32);(2,6)(7,8)(9,11,13,12)(10,14,15,16)(17,19,21,29)(18,23,24,26)(20,28,22,27)(25,32,30,31);(1,3,5,4)(2,7,6,8)(9,12,13,11)(10,14,15,16)(18,24)(19,27)(20,22)(23,31)(26,32)(28,29);(1,4,5,3)(2,7,6,8)(17,20,21,22)(18,25,24,30)(19,28,29,27)(23,32,26,31)Polycyclic presentation:
2^8|a^2=eh,b^2=f,c^2=ef,d^2=e,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×#26329, #6448175, #6448182, … |
| maximal normal subgroup of | C2×#26329, #6448175, #6448182, … |
| maximal quotient of | #360962, #363070, #365881, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 19317);