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 | #1999 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
437 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,3)(5,6)(7,12)(8,10)(9,11)(13,14,17,21)(15,22,25,19)(16,26,18,20)(23,28,24,27);(1,2,4,3)(5,7,9,8)(6,10,11,12)(13,15)(14,19)(16,24)(17,25)(18,23)(20,27)(21,22)(26,28);(1,2,4,3)(5,8,9,7)(6,10,11,12)(15,23)(16,18)(19,27)(20,26)(22,28)(24,25);(1,3,4,2)(6,11)(10,12)(13,16,17,18)(14,20,21,26)(15,24,25,23)(19,27,22,28)Polycyclic presentation:
2^8|a^2=h,b^2=eh,c^2=eh,d^2=fh,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,d^c=dh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#26815, #6443776, #6443784, … |
| maximal normal subgroup of | C2×#26815, #6443776, #6443784, … |
| maximal quotient of | #311343, #311344, #311351, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 19803);