Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23860 |
| 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 | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
460 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6,8,10)(7,11,12,9)(13,14)(15,23)(16,19)(17,27)(18,20)(21,25)(22,24)(26,28);(1,2)(3,4)(13,15,18,25)(14,19,20,28)(16,17,26,24)(21,27,23,22);(2,4)(5,7,8,12)(6,9,10,11)(14,20)(19,28)(21,23)(22,27);(13,16,18,26)(14,21,20,23)(15,24,25,17)(19,22,28,27)Polycyclic presentation:
2^8|a^2=fh,b^2=h,c^2=fh,d^2=h,e^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dh,e^b=eh,e^d=eh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#25981, #6438082, #6438084, … |
| maximal normal subgroup of | C2×#25981, #6438082, #6438084, … |
| maximal quotient of | #305405, #305407, #305432, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18969);