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) = C22×C4 |
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 ◁ C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
333 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)(4,5)(6,8)(9,10)(11,13)(12,14)(15,16)(17,18,21,25,20,24,30,23)(19,26,29,32,28,22,27,31);(2,5)(6,7)(9,11,12,15)(10,13,14,16)(17,19,21,29,20,28,30,27)(18,22,25,31,24,26,23,32);(10,14)(13,16)(18,23)(19,27)(21,30)(22,26)(24,25)(28,29);(1,3,4,8)(2,6,5,7)(18,24)(22,26)(23,25)(31,32)Polycyclic presentation:
2^8|a^2=g,b^2=f,d^2=eh,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,f^c=fh,g^c=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#27731, #6458421, #6458438, … |
| maximal normal subgroup of | C2×#27731, #6458421, #6458438, … |
| maximal quotient of | #360397, #362553, #366059, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 20719);