Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24264 |
| 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 | |
361 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);(1,3,4,8)(2,5,6,7)(17,18,21,25,22,26,20,24)(19,27,28,31,29,32,30,23);(9,11,12,16)(10,13,14,15)(17,19)(18,23)(20,28)(21,30)(22,29)(24,27)(25,32)(26,31);(2,6)(5,7)(17,20,22,21)(18,24,26,25)(19,28,29,30)(23,27,31,32)Polycyclic presentation:
2^8|a^2=f,b^2=eg,c^2=f,d^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,g^c=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#28773, #6459808, #6459826, … |
| maximal normal subgroup of | C2×#28773, #6459808, #6459826, … |
| maximal quotient of | #362135, #364332, #367438, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21761);