Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24274 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
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 | |
511 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,3,4)(5,6)(7,11)(8,10)(9,12)(13,14,18,15,17,21,19,22)(16,24,27,25,26,23,28,20);(5,7,8,12)(6,9,10,11)(13,15,18,21,17,22,19,14)(16,25,27,23,26,20,28,24);(2,4)(13,16,17,26)(14,20,21,25)(15,23,22,24)(18,28,19,27);(6,10)(9,11)(14,21)(15,22)(20,25)(23,24)Polycyclic presentation:
2^8|a^2=f,b^2=eg,c^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=dh,f^c=fh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#28355, #6459731, #6459740, … |
| maximal normal subgroup of | C2×#28355, #6459731, #6459740, … |
| maximal quotient of | #319345, #319361, #319376, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21343);