Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24274 |
| 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, prime radical | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
357 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6,8,9)(7,11,12,10)(13,14,16,19)(15,21,23,18)(17,20,25,27)(22,26,24,28);(6,9)(10,11)(13,15,17,24,16,23,25,22)(14,18,20,26,19,21,27,28);(1,2,3,4)(14,19)(15,22)(17,25)(18,26)(21,28)(23,24);(5,7,8,12)(6,10,9,11)(14,19)(15,23)(20,27)(22,24)Polycyclic presentation:
2^8|a^2=e,b^2=g,c^2=fh,d^2=eh,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=dh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#28652, #6464056, #6464062, … |
| maximal normal subgroup of | C2×#28652, #6464056, #6464062, … |
| maximal quotient of | #324999, #325000, #325003, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21640);