Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| 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×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
431 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)(11,15)(12,13)(14,16)(17,18,21,24)(19,22,27,31)(20,23,28,32)(25,29,26,30);(1,3,5,4)(2,6,8,7)(10,13)(14,15)(17,19)(18,22)(20,25)(21,27)(23,29)(24,31)(26,28)(30,32);(1,3,5,4)(2,7,8,6)(9,11,12,16)(10,14,13,15)(19,25)(20,28)(22,29)(23,32)(26,27)(30,31);(1,4,5,3)(2,7,8,6)(10,13)(14,15)Polycyclic presentation:
2^8|a^2=h,b^2=ef,c^2=e,d^2=ef,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#27526, #6440960, #6440978, … |
| maximal normal subgroup of | C2×#27526, #6440960, #6440978, … |
| maximal quotient of | #360294, #361545, #365146, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 20514);