Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30412 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #1999 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
404 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,8)(3,4,7,6)(9,10,12,14)(11,15,16,13)(18,21)(22,24);(1,3,5,7)(2,6,8,4)(9,11,12,16)(10,13,14,15)(18,22)(20,23)(21,24);(1,4,5,6)(2,7,8,3)(10,14)(13,15)(17,18,19,21)(20,24,23,22);(2,8)(3,7)(11,16)(13,15)Polycyclic presentation:
2^8|a^2=e,b^2=e,c^2=ef,g^2=h,b^a=be,c^a=cf,d^a=defh,c^b=cg,d^b=de,g^b=gh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #6637360, #6637375, #6637401, … |
| maximal normal subgroup of | #6637360, #6637375, #6637401, … |
| maximal quotient of | #384282, #384615, #384687, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 25050);