Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24378 |
| 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 ◁ C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
335 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)(3,7)(4,6)(5,8)(17,18,20,23,21,24,29,32)(19,25,27,30,26,31,28,22);(1,3,4,8)(2,5,6,7)(9,10)(11,14)(12,13)(15,16)(19,26)(22,30)(25,31)(27,28);(10,13)(14,16)(17,19,21,26)(18,22,24,30)(20,28,29,27)(23,31,32,25);(2,6)(5,7)(9,11,12,15)(10,14,13,16)Polycyclic presentation:
2^8|a^2=f,b^2=e,c^2=h,d^2=gh,f^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,f^c=fh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#29431, #6461597, #6461626, … |
| maximal normal subgroup of | C2×#29431, #6461597, #6461626, … |
| maximal quotient of | #363307, #364297, #371284, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22419);