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 | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #2414 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
471 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6)(7,11)(8,10)(9,12)(13,14)(15,19)(16,26)(17,22)(18,21)(20,23)(24,28)(25,27);(5,7,8,12)(6,9,10,11)(13,15,23,24,18,16,17,25)(14,19,20,28,21,26,22,27);(1,2,3,4)(13,16,18,15)(14,19,21,26)(17,25,23,24)(20,27,22,28);(6,10)(9,11)(13,17,18,23)(14,20,21,22)(15,24,16,25)(19,27,26,28)Polycyclic presentation:
2^8|b^2=egh,c^2=f,d^2=h,g^2=h,b^a=be,c^a=cf,d^a=deh,c^b=cg,d^b=dh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#28570, #6459490, #6459496, … |
| maximal normal subgroup of | C2×#28570, #6459490, #6459496, … |
| maximal quotient of | #324278, #324280, #324282, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21558);