Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24390 |
| 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 | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
339 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)(11,13,15,16)(17,18)(19,21)(20,27)(22,26)(23,30)(24,28)(25,31)(29,32);(1,3,5,4)(2,6,8,7)(9,11)(10,13)(12,15)(14,16)(17,19,21,18)(20,25,29,23)(22,26,28,24)(27,30,32,31);(1,3,5,4)(2,7,8,6)(11,15)(13,16)(17,20,28,31,21,29,22,30)(18,23,26,32,19,25,24,27);(1,4,5,3)(2,7,8,6)(18,19)(23,25)(24,26)(27,32)Polycyclic presentation:
2^8|a^2=g,b^2=e,c^2=ef,d^2=eh,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,d^b=dh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#27971, #6458391, #6458398, … |
| maximal normal subgroup of | C2×#27971, #6458391, #6458398, … |
| maximal quotient of | #360790, #361866, #365209, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 20959);