Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24264 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D8 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
440 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6)(7,11)(8,9)(10,12)(13,14)(15,21)(16,20)(17,24)(18,23)(19,25)(22,27)(26,28);(1,2)(3,4)(6,9)(10,11)(13,15,18,26,19,27,17,16)(14,20,24,22,25,28,23,21);(2,4)(13,16,17,27,19,26,18,15)(14,21,23,28,25,22,24,20);(5,7,8,12)(6,10,9,11)(13,16,17,27,19,26,18,15)(14,22,23,20,25,21,24,28)Polycyclic presentation:
2^8|b^2=f,c^2=fh,d^2=e,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,c^b=cg58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#26251, #6444493, #6444511, … |
| maximal normal subgroup of | C2×#26251, #6444493, #6444511, … |
| maximal quotient of | #307338, #307344, #307350, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 19239);