Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30396 |
| 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, prime radical | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
332 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,7)(3,4,8,6)(9,10)(11,15)(12,14)(13,16)(17,18,21,24)(19,22,26,30)(20,23,27,31)(25,29,28,32);(1,3,5,8)(2,6,7,4)(9,11)(10,13)(12,16)(14,15)(19,25)(20,27)(22,29)(23,31)(26,28)(30,32);(10,14)(13,15)(17,19,21,26)(18,22,24,30)(20,28,27,25)(23,32,31,29);(1,4,5,6)(2,3,7,8)(11,16)(13,15)Polycyclic presentation:
2^8|a^2=efh,b^2=ef,c^2=h,d^2=ef,g^2=h,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,g^b=gh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #6647789, #6647795, #6647797, … |
| maximal normal subgroup of | #6647789, #6647795, #6647797, … |
| maximal quotient of | #385068, #385073, #385490, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 25644);