Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29742 |
| 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 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
336 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,23)(20,24)(21,25)(22,27)(26,30)(28,32)(29,31);(1,2,4,5)(3,6,7,8)(17,19,26,29,20,27,21,28)(18,22,30,32,24,23,25,31);(1,3)(2,6)(4,7)(5,8)(10,13)(14,16)(17,19,20,27)(18,23,24,22)(21,29,26,28)(25,31,30,32);(3,7)(6,8)(9,11,12,15)(10,14,13,16)(19,27)(22,23)(28,29)(31,32)Polycyclic presentation:
2^8|b^2=eg,c^2=h,d^2=f,g^2=h,b^a=be,c^a=cf,c^b=cg,d^b=dh,d^c=de,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#30363, #6433931, #6433936, … |
| maximal normal subgroup of | C2×#30363, #6433931, #6433936, … |
| maximal quotient of | #380915, #380917, #380941, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23351);