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) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #2414 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
415 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)(17,18,22,21)(19,23,26,29)(20,24,28,30)(25,31,32,27);(1,3,4,8)(2,5,6,7)(10,13)(14,15)(17,19)(18,23)(20,27)(21,29)(22,26)(24,25)(28,31)(30,32);(9,11,12,16)(10,14,13,15)(17,20,22,28)(18,24,21,30)(19,25,26,32)(23,31,29,27);(2,6)(5,7)(10,13)(14,15)(17,18,22,21)(19,23,26,29)(20,24,28,30)(25,31,32,27)Polycyclic presentation:
2^8|a^2=h,b^2=ef,c^2=fh,d^2=h,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×#26209, #6441821, #6441840, … |
| maximal normal subgroup of | C2×#26209, #6441821, #6441840, … |
| maximal quotient of | #360891, #361292, #365793, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 19197);