Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C24 · Out(G) = C211 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C24 |
| commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
972 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10,12,14)(11,15,16,13)(17,18,20,22)(19,21,23,24);(2,5)(6,8)(17,19)(18,21)(20,23)(22,24);(9,11,12,16)(10,13,14,15)(18,22)(19,23);(19,23)(21,24);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^8|a^2=g,c^2=gh,e^2=f,b^a=bf,c^a=cg,c^b=ch,d^b=dh76 conjugacy classes — view the full character table.
| maximal subgroup of | #6385796, #6385798, #6385800, … |
| maximal normal subgroup of | #6385796, #6385798, #6385800, … |
| maximal quotient of | #514144, #514201, #514380, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27828);